yC-NRLF. 


7M7 


RUDIMENTS 


A?  T 


€ONT 


NHMEROUS    EXERCISES 


SLATE   AND    BTACKBOARD, 


FOR  BI1-- 


BY  OAMES  B.  iilOjViSON,  A.M., 

AUTHOR    OF    >*'  ,    EXERCISKS   IN   ARITn!ft<CTICAIi    ANfALYS 

PRACTICAL    ARITTIMKTIC  ;    UIGHKR    ARITHMETIC;    EDITOR    OF    DAV's 
SCHOOL   AL^KBK/t,    '.KG^NDRE's    GEOMSTRV>   ETC, 


oo 


CM 
O 


,W    YOKE: 

N  ,c  \TEY,  48    &   50  WALKER   ST. 

AGO :  S.  C.  G  RIGOS  &  CO.,  89  &  41  LAKT 

:vs  &  co.    ST.  LOUIS  :  KEITH  &  WOODS. 

•,.x. -23  &  CO.     DETROIT  I   EAYMOIO  &  SKLLiiOK. 
-.i.riMS.      BTJFFALO:   PH1NNKY    A   CO. 
.vaUSfr:    T.    S.    Q[JAOKENBU8H. 


. 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFOF 

GIFT  OF 


Received  .... 

Accession  No.  6  J  ^  ^  3~  -    Class  No. 


an&  ffifjomsan's   Series. 

RUDIMENTS 

OF 

AEITHMETIC; 


CONTAINING 


NUMEROUS    EXERCISES 


SI-ATE   AND    S&ACKBOARD, 


FOB  BEGINNERS 


BY  JAMES  IB.  THOMSON,  A.M., 
JSFf  ±P*    fllf   Tjfl  B         ^x^^1^^ 

AUTHOR  OK  MENTAL  ARiTHMETicr;  EXERCISES  IN  ARIT&MKTICAL  ANALYST** 

PRACTICAL  ARITHMETIC;  HtaH^fa ARITHMETIC;  E|<ITOR  OP  DA\  s 

SCHOOL   ALGEBRA;    LEGENT>RKySGEOMETR.V,   ETC. 


NEW    TOEK: 

IVISON  &  PHINNEY,  48    &  50  WALKER  ST. 
CHICAGO:  S.  C.  GEIGGS  &  CO.,  39  &  41  LAKE  ST. 

CINCINNATI  :   MOORE,  WIL8TACH,  KEYS  &  CO.     ST.   LOUIS  :   KEITH  &  WOODS. 

PHILADELPHIA;  SOWER,  BARNES  &  oo.    BUFFALO;  PHINNEY  fc  CO. 
NEWBUEQ:  T.  s.  QTTAOKENBUSH.  t 

1859. 


o 


Entered  according  to  Act  of  Congress,  in  the  year  1833,  Ifjr 

JAMES  B.  THOMSON, 
fa  the  Clerk's  Office  for  the  Southern  District  of  New  Yoft 


•TKREOTTPXD  BY  THOMAS   B.    81CITK, 
216  W1LLIAH  8TRKKT,  N.  V. 


PREFACE. 


EDUCATION,  in  its  comprehensive  sense,  is  the  business  of 
life.  The  exercises  of  the  school-room  lay  the  foundation ; 
the  superstructure  is  the  work  of  after  years.  If  these  exer- 
cises are  rightly  conducted,  the  pupil  oh  tains  the  rudiments 
of  science,  and  what  is  more  important,  he  learns  how  to 
study ,  how  to  think  and  reason,  and  is  thus  enabled  to  appro- 
priate the  means  of  knowledge  to  his  future  advancement. 
Any  system  of  instruction,  therefore,  which  does  not  embrace 
these  objects,  which  treats  a  child  as  a  mere  passive  recipient, 
is  palpably  defective.  It  is  destitute  of  some  of  the  most 
essential  means  of  mental  development,  and  is  calculated  to 
produce  pigmies,  instead  of  giant  intellects. 

The  question  is  often  asked,  "  What  is  the  best  method  of 
proceeding  with  pupils  commencing  the  study  of  Arithmetic, 
or  entering  upon  a  new  rule  ?" 

The  old  method. — Some  teachers  allow  every  pupil  to  cipher 
uon  his  own  hook;"  to  go  as  fast,  or  as  slow  as  he  pleases, 
without  reciting  a  single  example  or  rule,  or  stopping  to  in- 
quire the  "  why  and  the  wherefore"  of  a  single  operation. 
This  mode  of  teaching  is  a  relic  of  by-gone  days,  and  is  prima 
fade  evidence,  that  those  who  practice  it,  are  behind  the 
spirit  of  the  times. 

Another  method. — Others  who  admit  the  necessity  of  teach- 
ing arithmetic  in  classes,  send  their  pupils  to  their  seats,  and 
tell  them  to  u  study  the  rule."  After  idling  away  an  hour 
or  more,  up  goes  one  little  hand  after  another  with  the  de- 
spairing question : — "  Please  to  show  me  how  to  do  this  sum, 
sir  ?"  The  teacher  replies,  "  Study  the  rule ; — that  will  tell 
you."  At  length,  to  silence  their  increasing  importunity,  he 
takes  the  slate,  solves  the  question,  and,  without  a  word  of 


5V  PREFACE, 

explanation,  returns  it  to  its  owner.  He  thus  goes  through 
the  class.  When  the  hour  of  recitation  comes,  the  class  is 
not  prepared  with  the  lesson.  They  are  sent  to  their  seats 
to  make  another  trial,  which  results  in  no  "better  success. 
And  what  is  the  consequence?  They  are  discouraged  and 
disgusted  with  the  study. 

A  more  excellent  way. — Other  teachers  pursue  a  more  ex- 
cellent way,  especially  for  young  pupils.  It  is  this : — The 
teacher  reads  over  with  the  class  the  preliminary  explanations, 
and  after  satisfying  himself  that  they  understand  the  mean- 
ing of  the  terms,  he  calls  upon  one  to  read  and  analyze  the 
first  example,  and  set  it  down  upon  the  blackboard,  while 
the  rest  write  it  upon  their  slates.  The  one  at  the  bourd 
then  performs  the  operation  audibly,  and  those  with  their 
slates  follow  step  by  step. 

Another  is  now  called  to  the  board  and  requested  to  set 
down  the  second  example,  while  the  rest  write  the  same 
upon  their  slates,  and  solve  it  in  a  similar  manner.  He  then 
directs  them  to  take  the  third  example,  and  lets  them  try 
their  own  skill,  giving  each  such  aid  as  he  may  require.  In 
this  way  they  soon  get  hold  of  the  principle,  and  if  now  sent 
to  their  seats,  will  master  the  lesson  with  positive  delight. 

As  to  assistance,  no  specific  directions  can  he  given  which 
will  meet  every  case.  The  best  rule  is,  to  afford  the 
learner  just  that  kind  and  amount,  which  will  secure  the 
greatest  degree  of  exertion  on  his  part.  Less  than  this  dis- 
courages; more,  enervates. 

In  conclusion,  we  would  add,  that  this  elementary  work 
was  undertaken  at  the  particular  request  of  several  eminent 
practical  teachers,  and  is  designed  to  fill  a  niche  in  primary 
schools.  It  presents,  in  a  cheap  form,  a  series  of  progressive 
exercises  in  the  simple  and  compound  rules,  which  are 
adapted  to  the  capacities  of  beginners,  and  are  calculated  to 
form  habits  of  study,  awaken  the  attention,  and  strengthen 
the  intellect. 

J.  B.  THOMSON. 

KBW  YORK,  January,  1858. 


CONTENTS 


SECTION  I. 

ARITHMETIC  defined,         ....--••7 

Notation, 7 

Roman  Notation,  ......--7 

Arabic  Notation.          ......  -9 

Numeration, -12 

SECTION  II. 

ADDITION  defined, 16 

When  the  sum  of  a  column  does  not  exceed  9,  -  -  -18 
When  the  sura  of  a  column  exceeds  9,-  -  -  -  -19 
General  Rule  for  Addition,  -  -  -  -  -  -  -20 

SECTION  III. 

SUBTRACTION  defined, 27 

When  a  figure  in  the  lower  No.  is  smaller  than  that  above  it,  -  28 
When  a  figure  in  the  lower  No.  is  larger  than  that  above  it,  29 
Borrowing  10,  ....  30 

General  Rule  for  Subtraction,     -        -        -        ..        -81 

SECTION  IV. 

MULTIPLICATION  defined, 36 

When  the  multiplier  contains  but  one  figure,  89 

When  the  multiplier  contains  more  than  one  figure,  -  -  41 
General  Rule  for  Multiplication,  ....  43 

To  multiply  by  10,  100,  1000,  <fcc., 45 

When  there  are  ciphers  on  the  right  of  the  multiplier,  -  46 
When  there  are  ciphers  on  the  right  of  the  multiplicand,  -  47 
When  there  are  ciphers  on  the  right  of  both,  48 

SECTION  V. 
DIVISION  defined,     .......        .        .  49 

Short  Division,    -  52 


VI  CONTENTS. 

Rule  for  Short  Di  nsion, 54 

Long  Division, 56 

Difference  between  Short  and  Long  Division,        -        -        -    67 

Rule  for  Long  Division, 58 

To  dhide  by  10,  100,  1000,  <fec.,  -     61 

When  there  are  ciphers  on  the  right  of  the  divisor,  -        -         62 

SECTION  VI. 
FRACTIONS,    - 68 

To  find  what  part  one  given  number  is  of  another,  -        -        66 

A  part  of  a  number  being  given,  to  find  the  whole,  -         -     66 

To  multiply  a  whole  number  by  a  fraction,        -  -        -        67 

To  multiply  a  whole  by  a  mixed  number,     -        -  -        -     69 

To  divide  a  whole  number  by  a  fraction,  70 

To  divide  a  whole  by  a  mixed  number,          -        -  -        -71 

SECTION  VII. 

TABLES  in  Compound  Numbers,          -  •        -        -         74 

Paper  and  Books,          -        -         ..        .        .        .        -85 
Tables  of  aliquot  parts,     -------        87 

SECTION  VIII. 

ADDITION  of  Federal  Money, 90 

Subtraction  of  Federal  Money, 92 

Multiplication  of  Federal  Money, 93 

Division  of  Federal  Money, 94 

SECTION  IX. 

REDUCTION, 96 

Rule  for  Reduction  Descending, 97 

Rule  for  Reduction  Ascending, 100 

Compound  Addition, 106 

Compound  Subtraction, 108 

Compound  Multiplication, 110 

Compound  Division,      -        -        -        -        -         -        -        -11.1 

Miscellaneous  Exercises,  •        -        *         »        -        -        -       113 
Answers  to  Examples,  •  -        -        -  119 


ARITHMETIC. 


SECTION    I. 

ART,  !•  ARITHMETIC  is  the  science  of  numbers. 

Any  single  thing,  as  a  peach,  a  rose,  a  book,  is  called  a 
unit,  or  one ;  if  another  single  thing  is  put  with  it,  the 
collection  is  called  two  ;  if  another  still,  it  is  called  three ; 
if  another,  four  ;  if  another,  five,  &c. 

The  terms,  one,  tioo,  three,  four,  <kc.,  are  tke  names  of 
numbers.  Hence, 

2.  NUMBER  signifies  a  unit,  or  a  collection  of  units. 

Numbers  are  expressed  by  words,  by  letters,  and  bj 
figures. 

3*  NOTATION  is  the  art  of  expressing  numbers  by  letters 
or  figures.  There  are  two  methods  of  notation  in  use,  the 
Roman  and  the  Arabic. 

I.   ROMAN  NOTATION. 

4r»  The  Roman  Notation  is  the  method  of  expressing 
numbers  by  letters  ;  and  is  so  called  because  it  was  invented 
by  the  ancient  Romans.  It  employs  seven  capital  letters, 
viz  :  I,  V,  X,  L,  C,  D,  M. 

When  standing  alone,  the  letter  I,  denotes  one  ;  V,  fiv-e  ; 
X,  ten  ;  L,  fifty  ;  C,  one  hundred  ;  D,  five  hundred  ;  M, 
one  thousand. 


QUEST.— 1.  What  is  Arithmetic ?  What  is  a  single  thing  called?  If  an- 
other is  put  with  it,  what  is  the  collection  called?  If  another,  what  ?  What 
are  the  terras  one,  two,  three,  &c.  ?  2.  What  then  is  number  ?  How  are 
numbers  expressed  ?  3.  What  is  Notation  ?  How  many  methods  of  notation 
are  in  use?  4.  What  is  the  Roman  notation?  Why  so  called?  How  many 
»etters  does  it  employ?  What  does  the  letter  I,  denote?  V?  X?  L?  C?  D?  M? 


NOTATION.  !SECT.  i. 

5.  To  express  the  intervening  numbers  from  to  one  a 
thousand,  or  any  number  larger  than  a  thousand,  we  re* 
sort  to  repetitions  and  various  combinations  of  these  let- 
ters, as  may  be  seen  from  the  following 

TABLE. 


I         denotes 

one. 

XXXI 

denotes  thirty-one. 

II              " 

two. 

XL 

"     forty. 

111 

three. 

XLI 

"     forty-one. 

IV 

four. 

L 

"    fifty. 

V              " 

five. 

LI 

"     fifty-one. 

VI             « 

six. 

LX 

"     sixty. 

VII           « 

seven. 

LXI 

"    sixty-one. 

VIII 

eight. 

LXX 

"     seventy. 

IX            " 

nine. 

LXXX 

"     eighty. 

X             " 

ten. 

xc 

"    ninety. 

XI 

eleven. 

XCI 

"     ninety-one. 

XII           " 

twelve. 

C 

"     one  hundred. 

XIII 

thirteen. 

CI 

"     one  hund.  and  one 

XIV         " 

fourteen. 

CIV 

"     one  hund.  and  fout 

XV 

fifteen. 

ex 

"     one  hund.  and  ten. 

XVI          « 

sixteen. 

cc 

"     two  hundred. 

XVII        « 

seventeen. 

ccv 

"    two  hund.  and  five. 

XVIII      " 

eighteen. 

ccc 

"     three  hundred. 

XIX 

nineteen. 

cccc 

"     four  hundred. 

XX 

twenty. 

D 

"     five  hundred. 

XXI 

twenty-one. 

DC 

"     six  hundred. 

XXII        « 

twenty-two. 

DCC 

"     seven  hundred. 

XXIII      « 

twenty-three. 

DCCC 

"     eight  hundred. 

XXIV      " 

twenty-  four. 

DCCCC 

"     nine  hundred. 

XXV        " 

twenty-five. 

M 

"     one  thousand. 

XXVI       " 

twenty-six. 

MC 

"     one   thousand  and 

XXVII     " 

twenty-seven. 

one  hundred. 

XXVIII  " 

twenty-eight. 

MM 

"     two  thousand. 

XXIX      " 

twenty-nine. 

MDCCCL  "     one  thousand   eight 

XXX 

thirty. 

hundred  and  fifty. 

QUKST.— 5.  What  do  the  letters  IV,  denote?    VI?    VIII?    IX?   XI?    XI V» 
XVI?     XVTII?     XIX?     XXIV?    XL?     LXXX?     XC?     CIV?     Express 
seven  by  Id  icrs  on  the  slate  or  black-board.    How  express  eleven  ?    Thirteen 
Tweuty-flvof    Nineteen?    Forty-lour?    Eighty-seven?    Ninety-nine? 


ARTS.  5 — 7.]  NOTATION.  9 

OBS.  1.  Every  time  a  letter  is  repeated,  its  value  is  repeated. 
Thus,  the  letter  I,  standing  alone,  denotes  one  ;  II,  two  ones  or  two, 
<fcc.  So  X  denotes  ten  ;  XX,  twenty,  <fcc. 

2.  When  two  letters  of  different  value  are  joined  together,  if  the 
less  is  placed  before  the  greater,  the  value  of  the  greater  is  dimin- 
ished as  many  units  as  the  less  denotes ;  if  placed  after  the  greater, 
the  value  of  the  greater  is  increased  as  many  units  as  the  less  de- 
notes. Thus,  V  denotes  five ;  but  IV  denotes  only  four ;  and  VI, 
six.  So  X  denotes  ten ;  IX,  nine ;  XI,  eleven. 

Note. — The  questions  on  the  observations  may  be  omitted,  by 
beginners,  till  review,  if  deemed  advisable  by  the  teacher. 

II.   AKABIC  NOTATION. 

6.  The  Arabic  Notation  is  the  method  of  expressing 
numbers  by  figures  ;  and  is  so  called  because  it  is  supposed 
to  have  been  invented  by  the  Arabs.  It  employs  the  fol- 
lowing ten  characters  or  figures,  viz  : 

1234567890 

one,     two,     three,    four,     five,       six,     seven,    eight,    nine,    naught. 

OBS.  1.  The  first  nine  are  called  significant  figures,  because  each 
one  always  expresses  a  value,  or  denotes  some  number.  They  are 
also  called  digits,  from  the  Latin  word  digitus,  signifying  a  finger, 
because  the  ancients  used  to  count  on  their  fingers. 

2.  The  last  one  is  called  naught,  because  when  standing  alone, 
it  expresses  nothing,  or  the  absence  of  number.  It  is  also  called 
cipher  or  zero. 

7  •  All  numbers  larger  than  9,  are  expressed  by  different 
combinations  of  these  ten  figures.  For  example,  to  express 
ten,  we  use  the  1  and  0,  thus  10 ;  to  express  eleven,  we 
use  two  Is,  thus  11  ;  to  express  twelve,  we  use  the  1  and 
2,  thus  12,  <fec. 

QuKST.—Oi*.  What  is  the  effect  of  repeating  a  letter  ?  If  a  letter  of  lost 
value  is  placed  before  another  of  greater  value,  what  is  the  effect?  If  placed 
after,  what  ?  6.  What  is  the  Arabic  notation  ?  Why  so  called  ?  How  many 
figures  does  it  employ?  What  are  their  names?  Obs.  What  are  the  first  nino 
called  ?  Why  ?  What  else  are  they  sometimes  called?  What  is  the  last  one 
called?  Why?  7.  How  are  numbers  larger  than  nine  expressed ?  Express 
ten  by  figures.  Eleven.  Twelve.  Fifteen. 


10                                NOTATION.                    [SECT,  i 

Hie  method  of  expressing  numbers  by  figures  from 

one  to  a  thousand,  may  be  seen  from  the  following 

TABLE. 

1,  one. 

36,  thirty-six. 

71,  seventy-one. 

2,  two. 

37,  thirty-seven. 

72,  seventy-two. 

3,  three. 

38,  thirty-eight. 

73,  seventy-three. 

4,  four. 

39,  thirty-nine. 

74,  seventy-  four. 

5,  five. 

40,  forty. 

75,  seventy-five. 

6,  six. 

41,  forty-one. 

76,  seventy-six. 

7,  seven. 

42,  forty-two. 

77,  seventy-seven. 

8,  eight. 

43,  forty-three. 

78,  seventy-eight. 

9,  nine. 

44,  forty-  four. 

79,  seventy-nine. 

10,  ten. 

45,  forty-five. 

80,  eighty. 

11,  eleven. 

46,  forty-six. 

81,  eighty-one. 

12,  twelve. 

47,  forty-seven. 

82,  eighty-two. 

13,  thirteen. 

48,  forty-eight. 

83,  eighty-three. 

14,  fourteen. 

49,  forty-nine. 

84,  eighty-  four. 

15,  fifteen. 

50,  fifty. 

85,  eighty-five. 

16,  sixteen. 

51,  fifty-one. 

86,  eighty-six. 

17,  seventeen. 

52,  fifty-two. 

87,  eighty-seven 

18,  eighteen. 

53,  fifty-three. 

88,  eighty-eight. 

19,  nineteen. 

54,  fifty-four. 

89,  eighty-nine. 

20,  twenty. 

55,  fifty-five. 

90,  ninety. 

21,  twenty-one. 

56,  fifty-six. 

91,  ninety-one. 

22,  twen'y-two. 

57,  fifty-seven. 

92,  ninety-two. 

23,  twenty-three. 

58,  fifty-eight. 

93,  ninety-three. 

24,  twenty-  four. 

59,  fifty-nine. 

94,  ninety-four. 

25,  twenty-five. 

60,  sixty. 

95,  ninety-five.         / 

26,  twenty-six. 

61,  sixty-one. 

96,  ninety-six. 

27,  twenty-seven. 

62,  sixty-two. 

97,  ninety-seven 

28,  twenty-eight. 

63,  sixty-three. 

98,  ninety-eight. 

29,  twenty-nine. 

64,  sixty-  four. 

99,  ninety-nine. 

30,  thirty. 

65,  sixty-five. 

100,  one  hundred. 

31,  thirty-one. 

66,  sixty-six. 

200,  two  hundred. 

32,  thirty-two. 

67,  sixty-seven. 

300,  three  hundred 

33,  thirty-three. 

68,  sixty-eight. 

400,  four  hundred. 

34,  thirty-four. 

69,  sixty-nine. 

900,  nine  hundred. 

35.  thirty-five. 

70,  seventy. 

1000,  one  thousand. 

QUEST.— How  express  fifteen?  Twenty-five?  Forty-seven?  Thirty-six 
Seventy-three  One  hundred  and  one  ?  One  hundred  and  ten  ?  One  hundrev 
and  twdnty  T  Two  hundred  and  fifteen  ? 


ARTS.  8 — 11.]  NOTATION.  11 

8.  It  will  be  peiceived  from  the  foregoing  table,  that 
the  same  figures,  standing  in  different  places,  have  differ- 
ent values. 

When  they  stand  alone  or  in  the  right  hand  place,  they 
express  units  or  ones,  which  are  called  units  of  the  first 
order. 

When  they  stand  in  the  second  place,  they  express  tens, 
which  are  called  units  of  the  second  order. 

When  they  stand  in  the  third  place,  they  express  hun- 
dreds, which  are  called  units  of  the  third  order. 

When  they  stand  in  the  fourth  place,  they  express 
thousands,  which  are  called  units  of  the  fourth  order,  <fec. 

For  example,  the  figures  2,  3,  4,  and  5,  when  arranged 
thus,  2345,  denote  2  thousands,  3  hundreds,  4  tens,  and  5 
units  ;  when  arranged  thus,  5432,  they  denote  5  thousands, 
4  hundreds,  3  tens,  and  2  units. 

9»  Ten  units  make  one  ten,  ten  tens  make  one  hundred, 
and  ten  hundreds  make  one  thousand,  &c. ;  that  is,  ten  of 
any  lower  order,  are  equal  to  one  in  the  next  higher  order 
Hence,  universally, 

0 O.  Numbers  increase  from  right  to  left  in  a  tenfold 
ratio  ;  that  is,  each  removal  of  a  figure  one  place  towards 
the  left,  increases  its  value  ten  times. 

11.  The  different  values  which  the  same  figures  have, 
are  called  simple  and  local  values. 

The  simple  value  of  a  figure  is  the  value  which  it  ex- 
presses when  it  stands  alone,  or  in  the  right  hand  place. 

QUEST.— 8.  Do  the  same  figures  always  have  the  same  value  ?  When  stand- 
ing alone  or  in  the  right  hand  place,  what  do  they  express?  What  do  they 
express  when  standing  in  the  second  place?  In  the  third  place?  In  the 
fourth  ?  9.  How  many  units  make  one  ten  ?  How  many  tens  make  a  hun- 
dred ?  How  many  hundreds  make  a  thousand  ?  Generally,  how  many  of  any 
lower  order  are  required  to  make  one  of  the  next  higher  order  ?  10.  What  is 
the  general  law  by  which  numbers  increase  ?  What  is  the  effect  upon  the  value 
of  a  figure  to  remove  it  one  place  towards  the  left?  11.  What  are  the  differ* 
ent  values  of  the  same  figure  called  ?  What  is  the  simple  value  of  a  figure  ? 
What  the  local  value  ? 


12  NUMERATION.  [SECT.  1. 

The  simple  value  of  a  figure,  therefore,  is  the  numbel 
which  its  name  denotes. 

The  local  value  of  a  figure  is  the  increased  value  which 
:t  expresses  by  having  other  figures  placed  on  its  right. 
Hence,  the  local  value  of  a  figure  depends  on  its  locality, 
or  the  place  which  it  occupies  in  relation  to  other  num- 
bers with  which  it  is  connected.  (Art.  8.) 

OBS.  This  system  of  notation  is  also  called  the  decimal  system, 
because  numbers  increase  in  a  tenfold  ratio.  The  term  decimal  if 
derived  from  the  Latin  word  decem,  which  signifies  ten. 

NUMERATION. 

12*  The  art  of  reading  numbers  when  expressed  by 
figures,  is  called  Numeration. 


NUMERATION    TABLE. 


123       861       518       924      263 

Period  V.     Period  IV.     Period  III.      Period  II.       Period  I. 
Trillions.       Billions.          Millions.      Thousands.         Units. 

13.  The  different  orders  of  numbers  are  divided  into 
periods  of  three  figures  each,  'beginning  at  the  right  hand. 


QUEST.— Upon  what  does  the  local  value  of  a  figure  depend  ?  Obs.  What  ia 
this  system  of  notation  sometimes  called  ?  Why  ?  12.  What  is  Numeration  ? 
Repeat  the  numeration  table,  beginning  at  the  right  hand.  What  is  the  first 
place  on  the  right  called?  The  second  place?  The  third?  Fourth?  Fifth 
Sixth?  Seventh?  Eighth?  Ninth?  Tenth,  &c.?  13.  How  are  the  orders  of 
numbers  divided  ? 


ARTS.  12 — 14.]  NUMERATION.  13 

The  first,  or  right  hand  period  is  occupied  by  units,  tens, 
hundreds,  and  is  called  units'  period ;  the  second  is  oc- 
cupied by  thousands,  tens  of  thousands,  hundreds  of 
thousands,  and  is  called  thousands'  period,  &c. 

The  figures  in  the  table  are  read  thus :  One  hundred 
and  twenty-three  trillions,  eight  hundred  and  sixty-one 
billions,  five  hundred  and  eighteen  millions,  nine  hundred 
and  twenty-four  thousand,  two  hundred  and  sixty-three. 

1 4»  To  read  numbers  which  are  expressed  by  figures. 

Point  them  off  into  periods  of  three  figures  each  ;  then, 
beginning  at  the  left  hand,  read  the  figures  of  each  period 
as  though  it  stood  alone,  and  to  the  last  figure  of  each,  add 
the  name  of  the  period. 

OBS.  1.  The  learner  must  be  careful,  in  pointing  0$~  figures,  always 
to  begin  at  the  right  hand  ;  and  in  reading  them,  to  be^in  at  the 
left  hand. 

2.  Since  the  figures  in  the  first  or  right  hand  period  alw  lys  de- 
note units,  the  name  of  the  period  is  not  pronounced.  Hevxce,  in 
reading  figures,  when  no  period  is  mentioned,  it  is  always  u,ider- 
stood  to  be  the  right  hand,  or  units'  period. 

EXERCISES    IN    NUMERATION. 

Note. — At  first  the  pupil  should  be  required  to  apply  to  each  fig- 
ure the  name  of  the  place  which  it  occupies.  Thus,  beginning  at 
the  right  hand,  he  should  say,  "  Units,  tens,  hundreds,"  &c.,  and 
point  at  the  same  time  to  the  figure  standing  in  the  place  which  he 
mentions.  It  will  be  a  profitable  exercise  for  young  scholars  to 
write  the  examples  upon  their  slates  or  paper,  then  point  them  off 
into  periods,  and  read  them. 


QUEST.— What  is  the  first  period  called  ?  By  what  is  it  occupied  ?  What  is 
the  second  period  called?  By  what  occupied?  What  is  the  third  period 
called  ?  By  what  occupied  ?  Wnat  is  the  fourth  called  ?  By  what  occupied  f 
What  is  the  fifth  called?  By  what  occupied?  14.  How  do  you  read  nurabere 
expressed  by  figures  ?  Obs.  Where  begin  to  point  them  off?  Where  to  read 
them  ?  Do  you  pronounce  the  name  of  the  right  hand  periol  ?  When  no 
period  is  named,  what  is  understood  ? 


14 


NUMERATION. 


Read  the  following  numbers : 


[SECT.  I 


Ex.  1. 

97 

16. 

12642 

Si.      7620 

2. 

110 

17. 

20871 

32.      8040 

3. 

256 

18. 

17046 

33.      9638 

4. 

307 

19. 

43201 

34.     11000 

5. 

510 

20. 

80600 

35.     12100 

6. 

465 

21. 

4203 

36.     1402C 

7. 

1248 

22. 

65026 

37.     10001 

8. 

2381 

23. 

78007 

38.      5  020 

9. 

4026 

24. 

90210 

39.     18022 

10. 

6420 

25. 

5025 

40.     30401 

11. 

8600 

26. 

69008 

41.      2506 

12. 

7040 

27. 

100  000 

42.    402  321 

13. 

8000 

28. 

125  236 

43.     65  007 

14. 

9007 

29. 

6005 

44.    750  026 

15. 

10000 

30. 

462  400 

45.    804  420 

40. 

2325672 

50.       7289405287 

47. 

4502360 

51.      185205370000 

48. 

62840285 

52.     6423691450896 

49. 

425026951 

53.    75894128247625 

EXERCISES    IN    NOTATION. 

15»  To  express  numbers  by  figures. 

Begin  at  the  left  hand  of  the  highest  period,  and  write 
the  figures  of  each  period  as  though  it  stood  alone. 

If  any  intervening  order,  or  period  is  omitted  in  the 
given  number,  write  ciphers  in  its  place. 

Write  the  following  numbers  in  figures  upon  the  slate 
or  black-board. 

1.  Sixteen,  seventeen,  eighteen,  nineteen,  twenty. 

2.  Twenty-three,  twenty-five,  thirty,  thirty-three. 

3.  Forty-nine,  fifty-one,  sixty,  seventy-four. 

4.  Eighty-six,  ninety-three,  ninety- seven,  a  hundred. 

QUEST.— 15.  How  are  numbers  expressed  by  figures  ?    If  any  intervening 
order  is  omitted  in  the  example,  how  is  jts  place  supplied? 


ART.  15.J  NUMERATION.  19 

5.  One  hundred  and  ten. 

6.  Two  hundred  and  thirty-five. 

7.  Three  hundred  and  sixty. 

8.  Two  hundred  and  seven. 

9.  Four  hundred  and  eighty-one. 

10.  Six  hundred  and  ninety-seven. 

11.  One  thousand,  two  hundred  and  sixty-three. 

12.  Four  thousand,  seven  hundred  and  ninety-nine. 

13.  Sixty-five  thousand  and  three  hundred. 

14.  One  hundred  and  twelve  thousand,  six  hundred 
and  seventy-three. 

15.  Three  hundred  and  forty  thousand,  four  hundred 
and  eighty-five. 

16.  Two  millions,  five  hundred  and  sixty  thousand. 

17.  Eight  millions,  two  hundred  and  five  thousand, 
three  hundred  and  forty-five. 

18.  Ten  millions,  five  hundred  thousand,  six  hundred 
and  ninety-five. 

19.  Seventeen    millions,   six   hundred   and  forty-five 
thousand,  two  hundred  and  six. 

20.  Forty-one  millions,  six  hundred  and  twenty  thou- 
sand, one  hundred  and  twenty-six. 

21.  Twenty- two   millions,  six  hundred  thousand,  one 
hundred  and  forty-seven. 

22.  Three  hundred  and   sixty  millions,  nine  hundred 
and  fifty  thousand,  two  hundred  and  seventy. 

23.  Five  billions,  six  hundred  and  twenty-one  millions, 
seven  hundred  and  forty-seven  thousand,  nine  hundred 
and  fifty-four. 

24.  Thirty-seven  trillions,  four  hundred  and  sixty-three 
billions,  two  hundred  and  ninety-four  thousand,  fire  hun- 
dred and  seventy-two. 


1$  ADDITION.  [SECT.  IL 

SECTION   II. 

ADDITION. 

ART.  16.  Ex.  1.  Henry  paid  4  shillings  for  a  pair  of 
gloves,  7  shillings  for  a  cap,  and  2  shillings  for  a  knife : 
how  many  shillings  did  he  pay  for  all  ? 

Solution. — 4  shillings  and  7  shillings  are  11  shillings, 
and  2  shillings  are  13  shillings.  He  therefore  paid  13 
shillings  for  all. 

OBS.  The  preceding  operation  consists  in  finding  a  single  num- 
ber which  is  equal  to  the  several  given  numbers  united  together, 
and  is  called  Addition.  Hence, 

17»  ADDITION  is  the  process  of  uniting  two  or  more 
numbers  in  one  sum. 

The  answer,  or  number  obtained  by  addition,  is  called 
the  sum  or  amount. 

OBS.  When  the  numbers  to  be  added  are  all  of  the  same  kind,  or 
denomination,  the  operation  is  called  Simple  Addition. 

18»  Sign  of  Addition  (+).  The  sign  cf  addition  is 
a  perpendicular  cross  (  +  ),  called  plus,  and  shows  that 
the  numbers  between  which  it  is  placed,  are  to  be  added 
together.  Thus,  the  expression  6  +  8,  signifies  that  6  is 
to  be  added  to  8.  It  is  read,  "  6  plus  8,"  or <:  6  added  to  8." 

Note. — The  term  plus,  is  a  Latin  word,  orig'nally  signifying 
'*  rcore."  In  Arithmetic,  it  means  "  added  to." 


QUEST.— 17.  What  is  addition?  What  is  the  answer  called  ?  Obs.  When 
the  numbers  to  bo  added  are  all  o?  the  same  denomination,  what  is  the  ope- 
ration called?  18.  What  is  the  si^n  of  addition?  Who*  does  it  show  ?  JVWe. 
What  is  the  moaning  of  the  word  plus  ? 


ARTS.  16— 19.]  ADDITION.  17 

19.  Sign  of  Equality  (— ).  The  sign  of  equality  is 
two  horizontal  lines  (— ),  and  shows  that  the  numbers  be- 
tween which  it  is  placed,  are  equal  to  each  other.  Thus, 
the  expression  4  +  3  =  7,  denotes  that  4  added  to  3  are 
equal  to  7.  It  is  read,  "  4  plus  3  equal  7,"  or  "  the  sum 
cf  4  plus  3  is  equal  to  7."  18  +  5  =  7 

ADDITION  TABLE. 


2  and 

3  and 

4  and 

5  and 

1  are   3 

1  are   4 

1  are   5 

1  are   6 

2   "    4 

2   "    5 

2   "    6 

2  "    7 

3   "    5 

3   "    6 

3   "    7 

3   "   8 

4   "    6 

4   "   7 

4   "    8 

4   "    9 

5   "   7 

5   "   3 

5   "    9, 

5   "   10 

6   "    8 

6   "    9 

6   "   10 

6   "   11 

7   "   9 

7   "   10 

7   "   11 

7   "   12 

8   "   10 

8   "   11 

8   "   12 

8   "   13 

9   "   11 

9   "   12 

9   "   13 

9   "   14 

10   "   12 

10   "   13 

10   "   14 

10   "   15 

6  and 

7  and 

8  and 

9  and 

1  are   7 

1  are   8 

1  are   9 

1  are  10 

2   "   8 

2   "    9 

2   "   10 

2   "   11 

3   "   9 

3   "   10 

3   "   11 

3   "   12 

4   "   10 

4   "   11 

4   "   12 

4   "   13 

5   "   11 

5   "   12 

5   "   13 

5   "   14 

6   "   12 

6   "   13 

6   "   14 

6   "   15 

7   "   13 

7   "   14 

7   "   15 

7   "   16 

8   "   14 

8   "   15 

8   "   16 

8   "   17 

9   "   15 

9   "   16 

9   "   17 

9   "   18 

10   "   16 

10   "   17 

10   "   18 

10   "   19 

Note.— It  is  an  interesting  and  profitable  exercise  for  young  pupils 
to  recite  tables  in  concert.  But  it  will  not  do  to  depend  upon  this 
method  alone.  It  is  indispensable  for  every  scholar  who  desires  to 
De  accurate  either  in  arithmetic  or  business,  to  have  the  c  ommon 


QUKST.— 19.  What  is  the  sign  of  equality?    What  does  it  show? 
2 


18  ADDITION.  [SECT.  II. 


'  tab  es  distinctly  and  indelibly  fixed  in  his  mind.  Hence, 
after  a  taole  has  been  repeated  by  the  class  in  concert,  or  individ- 
ually, the  teacher  should  ask  many  promiscuous  questions,  to  prevent 
its  being  recited  mechanically,  from  a  knowledge  of  the  regular  in- 
crease of  numbers. 

EXAMPLES. 

!3O»   When  the  sum  of  a  column  does  not  exceed  9. 

Ex.  1.  George  gave  37  cents  for  his  Arithmetic,  and 
42  cents  for  his  Reader  :  how  many  cents  did  he  give  for 
both? 

Directions.  —  Write  the  numbers         Operation. 
under   each   other,  so   that  units       ^  & 
may  stand  under  units,  tens  under      jf  g 
tens,  and  draw  a  line  beneath  them.       3   7  price  of  Arith. 
Then,  beginning  at  the  right  hand       4   2     "     of  Read. 
or  units,  add  each  column  sepa-      - 
rately  in  the  following  manner  :  —       7   9  Ans. 
2  units  and  7  units  are  9  units.      Write  the  9  in  units 
place  under  the  column  added.     4  tens  and  3  tens  are 
Y  tens.     Write  the  7  in  tens'  place.     The  amount  is  79 
cents. 

Write  the  following  examples  upon  the  slate  or  black- 
board, and  find  the  sum  of  each  in  a  similar  manner  : 

(2.)  (3.)  (4.)  (5.) 

26  231  623  5734 

42  358  145  4253 

(6.)  (7.)  (8.)  (9.) 

425  3021  5120  3521 

132  1604  2403  1043 

321  2142  1375  4215 


10.  What  is  the  sum  of  4321  and  2475  ? 

11.  What  is  the  sum  of  32562  and  56214? 

12.  What  is  the  sum  of  521063  and  465725  ? 


ARTS.  20— 22.  J  ADDITION.  19 

21.  When  the  sum  of  a  column  exceeds  9. 

13.  A  merchant  sold  a  quantity  of  flour  for  458  dollars, 
a  quantity  of  tea  for  887  dollars,  and  sugar  for  689  dol- 
lars :  how  much  did  he  receive  for  all  ? 

Having  written  the  numbers  as  Operation. 

Defore,  we  proceed  thus:  9  units       458  price  of  flour, 
and  7  units  are  16  units,  and  8       887     "     of  tea. 
are  24  units,  or  we  may  simply       689     "     of  sugar, 
say  9  and  7  are  16,  and  8  are  24.     2034  dollars.  Ans. 
Now  24  is  equal  to  2  tens  and 

4  units.  We  therefore  set  the  4  units  or  right  hand  figure 
in  units'  place,  because  they  are  units  ;  and  reserving  the 
2  tens  or  left  hand  figure  in  the  mind,  add  it  to  the  column 
of  tens  because  it  is  tens.  Thus,  2  (which  was  reserved) 
and  8  are  10,  and  8  are  18,  and  5  are  23.  Set  the  3  or 
right  hand  figure  under  the  column  added,  and  reserving 
the  2  or  left  hand  figure  in  the  mind,  add  it  to  the  column 
of  hundreds,  because  it  is  hundreds.  Thus,  2  (which  was 
reserved)  and  6  are  8,  and  8  are  16,  and  4  are  20.  Set 
the  0  or  right  hand  figure  under  the  column  added  ;  and 
since  there  is  no  other  column  to  be  added,  write  the  2 
in  thousands'  place,  because  it  is  thousands. 

N.  B.  The  pupil  must  remember,  in  all  cases,  to  set  down  the 
whole  sum  of  the  last  or  l$ft  hand  column. 

22.  The  process  of  reserving  the  tens  or  left  hand  fig- 
ure, when  the  sum  of  a  column  exceeds  9,  and  adding  it 
mentally  to  the  next  column,  is  called  carrying  tens. 

Find  the  sum  of  each  of  the  following  examples  in  a 
similar  manner : 

(14.)  (15.)  (16.)  (17.) 

856  364  6502  8245 

764  488  497  4678 

1620  Ans.  602  8301  362 


20  ADDITION.  [SECT.  IL 

23.  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL  RULE  FOR  ADDITION". 

I.  Write  the  numbers  to  be  added  under  each  other,  so 
that  units  may  stand  under  units,  tens  under  tens,  &c. 

II.  Beginning  at  the  right  hand,  add  each  column  sepa- 
rately, and  if  the  sum  of  a  column  does  not  exceed  9,  write 
it  under  the  column  added.     But  if  the  sum  of  a  column 
exceeds  9,  write  the  units'  figure  under  the  column  and 
carry  the  tens  to  the  next  column. 

III.  Proceed  in  this  manner  through  all  the  orders,  and 
finally  set  down  the  whole  sum  of  the  last  or  left  hand 
column. 

24.  PROOF. — Beginning  at  the  top,  add  each  column 
downward,  and  if  the  second  result  is  the  same   as  the 
first,  the  work  is  supposed  to  be  rightt 

EXAMPLES  FOR  PRACTICE. 

(1.)  (2.)  (3.)  (4.) 

Pounds.  Feet.  Dollars  Yards. 

25  113  342  4608 

46  84  720  635 

_84  2_16  898  43 

(5.)  (6.)  (7.)  (8.) 

684  336  6387  8261 

948  859  593  387 

569  698  3045  13 

203  872  15  7^ 

9.  What  is  the  sum  of  46  inches  and  38  inches? 


QUEST. — 23.  How  do  you  write  numbers  for  addition?  When  tho  mim  of  a 
column  does  not  exceed  9,  how  proceed  ?  When  it  exceeds  9,  how  proceed  ? 
22.  What  is  meant  by  carrying  the  tens  ?  What  do  you  do  with  the  sum  of 
the  last  column  ?  24.  How  is  addition  proved  ? 


<  23, 24.]  ADDITION.  21 

10.  What  is  the  sum  of  51  feet  and  63  feet  ? 

11.  What  is  the  sum  of  75  dollars  and  93  dollars? 

%• 

12.  Add  together  45  rods,  63  rods,  and  84  rods. 

13.  Add  together  125  pounds,  231  pounds,  426  pounds. 

14.  Add  together  267  yards,  488  yards,  and  6 25  yards. 

15.  Henry  traveled  256  miles  by  steamboat  and  320 
miles  by  Railroad :  how  many  miles  did  he  travel  ? 

16.  George  met  two  droves  of  sheep ;  one  contained 
461,  and  the  other  375 :  how  many  sheep  were  there  in 
both  droves  ? 

17.  If  I  pay  230  dollars  for  a  horse,  and  385  dollars  for 
a  chaise,  how  much  shall  I  pay  for  both  ? 

18.  A  farmer  paid  85  dollars  for  a  yoke  of  oxen,  27 
dollars  for  a  cow,  and  69  dollars  for  a  horse :  how  much 
did  he  pay  for  all  ? 

19.  Find  the  sum  of  425,  346,  and  681. 

20.  Find  the  sum  of  135,  342,  and  778. 

21.  Find  the  sum  of  460,  845,  and  576. 

22.  Find  the  sum  of  2345,  4088,  and  260. 

23.  Find  the  sum  of  8990,  5632,  and  5863. 

24.  Find  the  sum  of  2842,  6361,  and  523. 

25.  Find  the  sum  of  602,  173,  586,  and  408. 

26.  Find  the  sum  of  424,  375,  626,  and  75. 

27.  Find  the  sum  of  24367,  61545,  and  20372. 

28.  Find  the  sum  of  43200,  72134,  and  56324. 

29.  A  young  man  paid  5  dollars  for  a  hat ;  6  dollars 
for  a  pair  of  boots,  27  dollars  for  a  suit  of  clothes,  and  19 
dollars  for  a  cloak :  how  much  did  he  pay  for  all  ? 

30.  A  man  paid  14  dollars  for  wood,  16  dollars  for  a 
stove,  and  28  dollars  for  coal :  how  many  dollars  did  he 
pay  for  all  ? 

31.  A  farmer  bought  a  plough  for  13  dollars,  a  cart 
for  46  dollars,  and  a  wagon  for  61  dollars  :  what  was  the 
price  of  all  ? 


22  ADDITION.  [SECT.  II 

32.  What  is  the  sum  of  261+31+256  +  17  ? 

33.  What  is  the  sum  of  163+478+82  +  19  ? 

34.  What  is  the  sum  of  428  +  632  +  76+394  ? 

35.  W3iat  is  the  sum  of  320  +  856  +  100+503? 

36.  What  is  the  sum  of  641+108  +  138+710  ? 

37.  What  is  the  sum  of  700  +  66  +  970+21  ? 

38.  What  is  the  sum  of  304+971+608+496  ? 

39.  What  is  the  sum  of  848  +  683+420+668  ? 

40.  What  is  the  sum  of  868+45  +  17+25+27+38? 

41.  What  is  the  sum  of  641  +  85+580+42+7+63  ? 

42.  What  is  the  sum  of  29+281+7+43+785+46? 

43.  A  farmer  sold  25  bushels  of  apples  to  one  man,  IT 
bushels  to  another,  45  bushels  to  another,  and  63  bushels 
to  another :  how  many  bushels  did  he  sell  ? 

44.  A  merchant  bought  one  piece  of  cloth  containing 
25  yards,  another  28  yards,  another  34  yards,  and  an- 
other 46  yards :  how  many  yards  did  he  buy  ? 

45.  A  man  bought  3  farms ;  one  contained  120  acres, 
another  246  acres,  and  the  other  365  acres :  how  many 
acres  did  they  all  contain  ? 

46.  A  traveler  met  four  droves  of  cattle ;  the  first  con- 
tained 260,  the  second  175,  the  third  342,  and  the  fourth 
420 :  hOw  many  cattle  did  the  four  droves  contain  ? 

47.  A  carpenter  built  one  house  for  2365  dollars,  an- 
other for  1648  dollars,  another  for  3281  dollars,  and  an- 
other for  5260  dollars :  how  much  did  he  receive  for  all9 

48.  Find  the  sum  of  six  hundred  and  fifty- four,  eighty- 
nine,  four  hundred  and  sixty-three,  and  seventy- six. 

49.  Find  the  sum  of  two  thousand  and  forty-seven, 
three  hundred  and  forty-five,  thirty-six,  and  one  hundred. 

50.  In  January  there  are  31  days,  February  28,  March 
81,  April  30,  May  31,  June  30,  July  31,  August  31, .Sep- 
tember 30,  October  31,  November  30,  and  December  31 : 
how  many  days  are  there  in  a  year  ? 


ART.  24.a.]  ADDITION.  23 

24r«a.  Accuracy  and  rapidity  in  adding  can  be  ac- 
quired only  by  practice.  The  following  exercises  are  de- 
signed to  secure  this  important  object. 

OBS.  1.  In  solving  the  following  examples,  it  is  recommended 
to  the  pupil  simply  to  pronounce  the  result,  as  he  adds  each  suc- 
cessive figure.  Thus,  in  Ex.  1,  instead  of  saying  2  and  2  are  4, 
and  2  are  6,  &/c.,  proceed  in  the  following  manner :  "  two,  four,  six; 
eight,  ten,  twelve,  fourteen,  sixteen,  eighteen,  twenty."  Set  down 
naught  and  carry  two.  "  Two,  (to  carry)  three,  six,  nine,"  &c. 

2.  When  two  or  three  figures  taken  together  make  10,  as  8  and  2, 
7  and  3,  or  2,  3,  and  5,  k  accelerates  the  process  to  add  their  sum 
at  once.  Thus,  in  Ex.  4,  the  pupil  should  say:  "  ten  (1+9),  six- 
teen (6),  twenty-six  (5+5),  thirty-six  (2+8),"  &c. 

(1.)  (2.)  (3.)  (4.) 

32  654  987  463 

32  654  987  647 

32  654  987  455 

32  654  987  258 

32  654  987  572 

32  654  987  595 

32  654  987  615 

32  654  987  346 

32  654  987  729 

12  114  117  181 

(5.)  (6.)  (7.)  (8.) 

614  2140  8675  9244 

452  8963  2433  1432 

528  1232  6182  7234 

539  7855  2921  2523 

420  2123  2209  8440 

385  3333  4863  4346 

355  7674  6558  6704 

134  4521  5434  1852 

976  6589  5276  9258 

468  2637  8789  8106 


24 


ADDITION. 


[SECT.  IL 


(9.) 

(10.) 

.  (11.) 

(12.) 

4360 

9201 

42671 

62125 

7046 

7283 

68439 

31684 

5724 

4627 

32074 

22435 

5385 

6436 

47616 

16725 

8275 

9874 

30045 

94381 

9342 

8400 

26765 

25036 

6768 

6645 

10850 

85474 

5020 

4365 

25232 

10325 

9384 

8640 

43679 

42312 

(13.) 

(14.) 

(15.) 

(16.) 

2720 

5764 

27856 

47639 

4382 

5346 

32534 

23421 

2640 

3042 

20631 

34323 

3047 

5268 

34327 

71036 

2163 

3161 

53102 

62342 

6741 

2560 

92763 

57654 

1360 

7304 

51834 

32103 

7056 

2723 

23452 

53728 

8554 

8459 

62327 

61342 

4275 

6715 

50632 

23201 

(17.) 

(18.) 

(19.) 

(20.) 

4521 

6845 

75360 

89537 

3432 

3151 

27838 

23264 

4327 

2327 

42627 

41728 

6238 

4235 

34872 

74263 

5494 

2835 

63538 

21031 

3217 

5473 

54321 

53426 

2382 

9864 

63054 

91342 

4723 

3103 

29872 

23465 

3604 

7382 

63541 

38754 

2352 

5461 

53279 

94642 

ADDITION. 


(21.) 

(22.) 

(23.) 

(24.) 

8564 

56,430 

84,703 

341,725 

4736 

31,932 

19,384 

227,265 

3405 

29,754 

21,705 

311,265 

§037 

46,536 

43,641 

200,378 

6571 

86,075 

27,469 

421,850 

7439 

30,235 

52,267 

370,432 

4525 

41,623 

61,383 

174,370 

3137 

45,810 

75,604 

831,031 

2743 

56,239 

43,876 

580,456 

(25.) 

(26.) 

(27.) 

(28.) 

7243 

31,625 

68,901 

460,732 

2034 

51,482 

50,345 

804,045 

3710 

49,061 

75,005 

346,325 

5634 

80,604 

29,450 

450,673 

1730 

24,540 

80,063 

859,721 

5613 

67,239 

91,700 

236,548 

3005 

24,307 

43,621 

632,462 

7206 

58,392 

47,834 

753,324 

4354 

70,300 

83,276 

970,300 

7821 

56,749 

25,327 

267,436 

(29.) 

(30.) 

(31.) 

(32.) 

6458 

75,340 

64,268 

346,768 

2435 

6,731 

405 

21,380 

4678 

748 

1,708 

4,075 

4962 

68,451 

43,671 

126,849 

5143 

396 

72,049 

257 

8437 

7,503 

492 

L305 

7643 

46,075 

1,760 

24,350 

6850 

1,290 

25,357 

439,871 

7063 

25,738 

1,434 

40,306 

8324 

46,803 

84,162 

601,734 

26 


ADDITION. 


[SECT.  II, 


(S3.) 

(34.) 

(35.) 

(36.) 

423,674 

632,153 

317,232 

412,783 

307,316 

420,432 

203,671 

631,432 

730,248 

323,680 

334,263 

572,316 

506,213 

507,325 

210,600 

231,254 

110,897 

383,734 

356,237 

673,323 

206,341 

634,156 

264,871 

217,067 

324,563 

450,071 

531,634 

306,421 

185,174 

803,463 

342,106 

764,315 

364,230 

160,705 

768,342 

207,254 

150,176 

300,430 

407,821 

843,552 

843,204 

461,007 

311,289 

321,634 

370,679 

297,313 

564,735 

502,543 

445,168 

813,792 

470,334 

617,405 

370,432 

200,406 

436,216 

506,032 

5,338,315^4? 

is.  6,388,667 

Ans.  621,353 

762,573 

(37.) 

(38.) 

(39.) 

(40.) 

674,326 

783,457 

863,725 

958,439 

453,403 

675,306 

755,387 

843,670 

561,734 

858,642 

964,845 

784,561 

789,867 

246,468 

836,450 

976,435 

645,275 

587,649 

645,265 

833,406 

576,182 

523,731 

783,842 

797,624 

934,922 

445,372 

532,653 

845,358 

423,641 

832,148 

647,412 

978,262 

561,232 

465,363 

481,735 

784,643 

143,671 

642,742 

824,364 

865,343 

238,406 

830,423 

537,572 

976,736 

453,762 

256,372 

463,489 

853,974 

984,651 

662,456 

827,343 

467,852 

845,359 

572,834 

642,536 

948,685 

967,423 

864,213 

725,342 

896,872 

A.RTS.  25       27.]  SUBTRACTION.  2T 

SECTION  III. 

SUBTRACTION. 

ART.  25.  Ex.  1.  Charles  having  15  cents,  gave  6  cents 
for  an  orange :  how  many  cents  did  he  have  left  ? 

Solution. — 6  cents  taken  from  15  cents  leave  9  cents. 
Therefore  he  had  9  cents  left. 

OBS.  The  preceding  operation  consists  in  taking  a  less  number 
from  a  greater,  and  is  called  Subtraction.  Hence, 

26.  SUBTRACTION  is  the  process  of  finding  the  differ- 
ence between  two  numbers. 

The  answer,  or  number  obtained  by  subtraction,  is  called 
the  difference  or  remainder. 

OBS.  1.  The  number  to  be  subtracted  is  often  called  the  subtra- 
hend, and  the  number  from  which  it  is  subtracted,  the  minuend. 
These  terms,  however,  are  calculated  to  embarrass,  rather  than 
assist  the  learner,  and  are  properly  falling  into  disuse. 

2.  When  the  given  numbers  are  all  of  the  same  kind,  or  denomi- 
nation, the  operation  is  called  Simple  Subtraction. 

27.  Sign  of  Subtraction  (— ).     The  sign  of  subtrac- 
tion is  a  horizontal  line  (—•),  called  minus,  and  shows 
that  the  number  after  it  is  to  be  subtracted  from  the  one 
before  it.     Thus  the  expression  7 —  3,  signifies  that  3  is  to  be 
subtracted  from  7  ;  and  is  read,  "  7  minus  3,"  or  "  7  less  3." 
Bead  the  following:   18—7  =  20  —  9.     23  —  10=16  —  3 
35  —  8  =  31  —  4. 

Note. — The  term  minus  is  a  Latin  word  signifying  less. 

QUEST.— C26.  What  is  subtraction?  What  is  the  answer  called?  Obs. 
What  is  the  number  to  be  subtracted  sometimes  called  ?  That  from  which  it 
Is  subtracted  ?  When  tho  given  numbers  are  of  the  same  denomination,  what 
is  the  operation  called  ?  27.  What  is  the  sign  of  subtraction  ?  What  does  it 
ihow  1  Note.  What  is  t<he  moaning  of  the  term  minus? 


SUBTRACTION. 


[SECT,  ill 


SUBTRACTION   TABLE. 


2  from 

3  from 

4  from 

5  from 

2  leaves    0 

3  leaves    0 

4  leaves    0 

5  leaves   0 

3      "        1 

4      "        1 

5      "        1 

6      "        1 

4      "        2 

5      "        2 

6      "        2 

7      "        2 

5      "        3 

6      "        3 

7      "        3 

8      "        3 

6      "        4 

7      "       4 

8      «        4 

9      "        4 

7      "        5 

8      "        5 

9      "        5 

10      "        5 

8      «        6 

9      "        6 

10      "        6 

11      "        6 

9      "        7 

10      "        7 

11      "        7 

12      "        7 

10      "        8 

11      "        8 

12      "        8 

13      "        8 

11      "        9 

12      "        9 

13      "        9 

14      «        9 

12      "      10 

13      "      10 

14      "      10 

15      "      10 

6  from 

7  from 

8  from 

9  from 

6  leaves    0 

7  leaves    0 

8  leaves    0 

9  leaves    0 

7      "        1 

8      "        1 

9      "        1 

10      "        1 

8      "        2 

9      "        2 

10      "        2 

11      "        2 

9      "        3 

10      "        3 

11      "        3 

12      "        3 

10      "        4 

11      "        4 

12      "        4 

13      "        4 

11      "        5 

12      "        5 

13      "        5 

14      "        5 

12      "        6 

13      "        6 

14      "        6 

15      "        6 

13      "        7 

14      "        7 

15      "        7 

16      "        7 

14      "        8 

15      "        8 

16      "        8 

17      •'        8 

15      "        9 

16      "        9 

17      "        9 

18      "        9  . 

16      "      10 

17      "      10 

18      "      10 

19      "      10  1 

OBS.  This  Table  is  the  reverse  of  Addition  Table.  Hence,  if  the 
pupil  has  thoroughly  learned  that,  it  will  cost  him  but  little  time  or 
trouble  to  learn  this.  (See  observations  under  Addition  Table.) 

EXAMPLES. 

28.  When  each  figure  in  the  lower  number  is  smaller 
than  the  figure  above  it. 

1.  A  farmer  raised  257  bushels  of  apples,  and  123 
bushels  of  pears :  how  many  more  apples  did  he  rais« 
than  pears  ? 


ARTS.  28,  29.]          SUBTRACTION.  29 

Directions. — Write     the     less  Operation. 

number    under    the  greater,   so 
that  units  may  stand  under  units,  *g    ^  jg 

tens  mder  tos,  &c.,  and  draw  a  Jj   _§    § 

line    beneath   them.     Beginning  257  apples, 

with  the  units  or  right  hand  fig-  123  pears, 

ure,  subtract  each  figure  in  the     JRem.    134  bush. 
»ower  numDor  from   the   figure 

above  it,  in  the  following  manner :  3  units  from  7  units 
leave  4  units.  Write  the  4  in  units'  place  under  the 
figure  subtracted.  2  tens  from  5  tens  leave  3  tens ;  set 
3  in  tens'  place.  1  hundred  from  2  hundred  leaves  1  hun- 
dred ;  write  the  1  hundred  in  hundreds'  place. 

Solve  the  following  examples  in  a  similar  manner : 

(2.)  (3.)  (4.)  (5.) 

From  45  68  276  698 

Take  JU  ££  123  453 

(6.)  (7.)  (8.)  (9.) 

From  54  delis.        76  pounds.       257  yds.       325  shil. 
Take   Cd  dolls.        64  pounds.        142  yds.        103  shil. 

10.  Samuel  having  436  marbles,  lost   214  of  them: 
how  many  had  he  left  ? 

29»   When  a  figure  in  the  lower  number  is  larger  than 
the  figure  above  it. 

11.  A  man  bought  63  bushels  of  wheat,  and  after- 
wards sold  37:  how  many  bushels  had  he  left? 

It  is    bv>  jus  tb'.tt  we  cannot  take  7      Ifirst  Method. 
units  fi  <    t  3  UIP',S,  for  7  is  larger  than  63 

3  ;  we  tl  erefove  add  10  to  the  3  units,  37 

and  it  w  U  raake  13  units ;  then  7  from      Rem.  26  bu. 
13  leave  i  6  ;  write  the  6  in  units'  place 
under  the  figure  subtracted.     To  compensate  for  the  10 


30  SUBTRACTION.  (SECT      [11 

units  we  added  to  the  upper  figure,  we  add  1  ten  to  the 

3  tens  or  next  figure  in  the  lower  number,  and  it  makes 

4  tens ;  and  4  tens  from  6  tens  leave  2  tens :  write  the  2 
in  tens'  place  Ans.  26  bushels. 

We  may  also  illustrate  the  process  of  borrowing  in  tfec 
following  manner : 

63  is  composed  of  6  tens  and  3         Second  Method. 
units.     Taking  1  ten  from  6  tens,          63  =  50  +  13* 
and  adding  it  to  the  3  units,  we          37=30+7 
have63  =  50+13.  Separating  the     JRem.  =  2Q  +   6,  or  26 
lower  number  into  tens  and  units, 

we  have  37  =  30  +  7.  Now,  substracting  as  before,  7 
from  13  leaves  6.  Then  as  we  took  1  ten  from  the  6  tens, 
we  have  but  5  tens  left ;  and  3  tens  from  5  tens  leave  2 
tens.  The  remainder  is  26,  the  same  as  before. 

3O»  The  process  of  taking  one  from  a  higher  order  in 
the  upper  number,  and  adding  it  to  the  figure  from  which 
the  subtraction  is  to  be  made,  is  called  borrowing  ten,  and 
is  the  reverse  of  carrying  ten.  (Art.  22.) 

OBS.  When  we  borrow  ten  we  must  always  remember  to  pay  it 
This  may  be  done,  as  we  have  just  seen,  either  by  adding  1  to  tte 
next  figure  in  the  lower  number,  or  by  considering  the  nextjigur* 
in  the  upper  number  1  less  than  it  is . 

12.  From  240  subtract  134,  and  prove  the  operation. 

Since  4  cannot  be  taken  from  0,  we  Operation. 

borrow  10;  then  4  from  10  leaves  6.     1  240 

added  to  3  (to  compensate  for  the  10  we  134 
borrowed)  makes  4,  and  4  from  4  leaves  0.       106  Ans. 
1  from  2  leaves  1. 

PROOF. — We  add  the  remainder  Proof. 

to  the  smaller  number,  and  since  the  134  less  No. 

sum  is  equal  to  the  larger  number,  106  remainder, 

the  work  is  right.  240  greater  No. 


ARTS.  30 — 32.  j          SUBTRACTION.  SI 

Solve  the  following  examples,  and  prove  the  operation. 

(13.)  (14.)  (15.)  (16.) 

From  375  5273  6474  8650 

Take_238  2657  3204  5447 

17.  From  8461875, take  3096208. 

31»  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL   RULE   FOR   SUBTRACTION. 

I.  Write  the  less  number  under  the  greater,  so  that  units 
may  stand  under  units,  tens  under  tens,  &c. 

II.  Beginning  at  the  right  hand,  subtract  each  figure  in 
the  lower  number  from  the  figure  above  it,  and  set  the  re- 
mainder under  the  figure  subtracted. 

III.  When  a  figure  in  the  lower  number  is  larger  than 
that  above  it,  add  10  to  the  upper  figure  ;  then  subtract  as 
before,  and  add  1  to  the  next  figure  in  the  lower  number. 

32*  PROOF. — Add  the  remainder  to  the  smaller  num- 
ber ;  and  if  the  sum  is  equal  to  the  larger  number,  the  work 
is  right. 

OBS.  This  method  of  proof  depends  upon  the  obvious  principle, 
that  if  the  difference  between  two  numbers  be  added  to  the  less,  the 
sum  must  be  equal  to  the  greater. 

EXAMPLES    FOR.  PRACTICE. 


(1.) 

From  325 

(2.) 
431 

(3.) 
562 

(4.) 
600 

Take  108 

249 

320 

231 

(5.) 
From  2230 

(6.) 

3042 

(*•) 

6500 

(8.) 
8435 

Take  1201 

2034 

3211 

5001 

QUEST. — 31.  How  do  you  write  numbers  for  subtraction  ?  Where  do  yon 
begin  to  subtract  ?  When  a  figure  in  the  lower  number  is  larger  than  the  one 
ibovo  it,  how  do  you  proceed  ?  32,  How  is  subtraction  pro\*xl  ? 


SUBTRACTION.  [SEOT.    Ill 


(11.) 

From  45100  826340  1000000 

Take   10000  513683  999999 

12.  From  132  dollars  subtract  109  dollars. 

13.  From  142  bushels  subtract  85  bushels. 

14.  From  375  pounds  subtract  100  pounds. 

15.  From  698  yards  subtract  85  yards. 

16.  From  485  rods  subtract  175  rods. 

17.  Take  230  gallons  from  460  gallons. 

18.  Take  168  hogsheads  from  671  hogsheads. 

19.  Take  192  bushels  from  268  bushels. 

20.  From  275  dollars  take  148  dollars. 

21.  From  468  pounds  take  219  pounds. 

22.  From  3246  rods  take  2164  rods. 

23.  From  45216  take  32200. 

24.  From  871410  take  560642. 

25.  From  926500  take  462126. 

26.  From  6284678  take  1040640. 

27.  468—423.  37.  17265—13167. 

28.  675—367.  38.  21480—20372. 

29.  800—560.  39.  30671—26140. 

30.  701—643.  40.  45723—31203, 

31.  963—421.  41.   81647—57025. 

32.  3263—1242.  42.  265328—140300. 

33.  4165—2340.  43.  170643—106340. 

34.  5600—3000.  44.  465746—241680. 

35.  7246—4161.  45.  694270—  590S95. 

36.  8670—7364.  46.  920486—500000. 

47.  A  man  having  235  sheep,  lost  163  of  them:  ho* 
many  had  he  left  ? 

48.  A  farmer  having  500  bushels  of  wheat,  sold  278 
bushels  :  how  much  wheat  had  he  left  ? 

49.  A  man  paid  625  dollars  for  a  carriage  and  430 


ART.  32.]  SUBTRACTION.  33 

dollars  for  a  span  of  horses :  how  much  more  did  he  pay 
for  his  carriage  than  for  his  horses  ? 

50.  A  man  gave  1263  dollars  for  a  lot,  and  2385  dol- 
lars for  building  a  house :  how  much  more  did  his  house 
cost  than  his  lot  ? 

51.  If  a  person  has  3290  dollars  in  real  estate,  and 
owes  1631  dollars,  how  much  is  he  worth? 

52.  A  man  gave  his  son  8263  dollars,  and  his  daughter 
5240  dollars :  how  much  more  did  he  give  his  son  tha» 
his  daughter? 

53.  A  man  bought  a  farm  for  9467  dollars,  and  sold 
it  for  11230  dollars :  how  much  did  he  make  by  his  bar- 
gain ? 

54.  If  a  man's  income  is  10000  dollars  a  year,  and  his 
expenses  6253  dollars,  how  much  will  he  lay  up  ? 

55.  The  captain  of  a  ship  having  a  cargo  of  goods 
worth  29230  dollars,  threw  overboard  in  a  storm  13216 
dollars'  worth :  what  was  the  value  of  the  goods  left  ? 

56.  A  merchant  bought  a  quantity  of  goods  for  12645 
dollars,  and  afterwards  sold   them  for   13960    dollars: 
how  much  did  he  gain  by  his  bargain  ? 

57.  A  man  paid  23645  dollars  fora  ship  and  after- 
wards  sold  it  for  18260  dollars  :  how  much  did  he  lose 
by  his  bargain  ? 

58.  The  salary  of  the  President  of  the  United  States  is 
25000  dollars  a  year  ;  now  if  his  expenses  are  19265  dol- 
lars, how  much  will  he  lay  up  ? 

59.  A  general  before  commencing  a  battle,  had  35260 
soldiers  in  his  army  ;  after  the  battle  he  had  only  21316: 
how  many  soldiers  did  he  lose  ? 

60.  The  distance  of  the  sun  from  the  earth  is  95000000 
miles ;  the  distance  of  the  moon  from  the  earth  is  240000 
miles :  how  much  farther  from  the  earth  is  the  sun  than 
the  moon  ? 

2 


34  SUBTRACTION.  [SECT.  Ill 

EXAMPLES    INVOLVING    ADDITION    AND    SUBTRACTION. 

61.  Henry  bought  63  oranges  of  one  grocer,  and  26 
of  another;  he  afterwards  sold  72:    how  many  oranges 
did  he  have  left  ? 

62.  Charles  had  47  marbles,  and  his  father  gave  him 
36  more;  he  afterwards  lost  50:  how  many  marbles  did 
he  then  have  ? 

63.  A  farmer  having   158  sheep,  lost  30  of  them  by 
sickness  and  sold  52 :  how  many  sheep  did  he  have  left? 

64.  Sarah's  father  gave  her  60  cents,  and  her  mother 
gave  her  54  cents ;  if  she  spends  62  cents  for  a  pair  of 
gloves,  how  many  cents  will  she  have  left  ? 

65.  A  merchant  purchased  a  piece  of  silk  containing 
78  yards;  he  then  sold  18  yards  to  one  lady,  and  17  to 
another :  how  many  yards  had  he  left  ? 

66.  If  a  man  has  property  in  his  possession  worth 
215   dollars,   and  owes   39   dollars   to  one  person,  and 
54  dollars  to  another,  how  much  money  will  he  have  left, 
when  he  pays  his  debts  ? 

67.  If  a  man's  income  is  185  dollars  per  month,  and 
he  pays  35  dollars  for  house  rent,  and  63  dollars  for  pro- 
visions per  month,  how  many  dollars  will  he  have  left  for 
other  expenses  ? 

68.  George  having  74  pears,  gave  away  43  of  them ; 
if  he  should  buy  35  more,  how  many  would  he  then 
have? 

69.  If  you  add  115  to  78,  and  from  the  sum  take  134, 
•what  will  the  remainder  be  ? 

70.  If  you  subtract  93  from  147,  and  add  110  to  the 
remainder,  what  will  the  sum  be  ? 

71.  A  merchant  purchased  125  pounds  of  butter  of 
one  dairy-man,  and  187  pounds  of  another ;  he  afterwards 
sold  163  pounds:  how  many  pounds  did  he  have  left? 


ART.  32.]  SUBTRACTION.  35 

72.  A  miller  bought  200  bushels  of  wheat  of  one 
farmer,  and  153  bushels  of  another;  he  afterwards  sold 
180  bushels :  how  many  bushels  did  he  have  left? 

73.  A  man  traveled  538  miles  in  3  days  ;  the  first  day 
he  traveled  149  miles,  the  second  day,  126  miles  :  how 
far  did  he  travel  the  third  day  ? 

74.  A  grocer  bought  a  cask  of  oil  containing  256  gal- 
lons ;  after  selling  93  gallons,  he  perceived  the  cask  was 
leaky,  and  on  measuring  what  was  left,  found  he  had  38 
gallons :  how  many  gallons  had  leaked  out  ? 

75.  A  manufacturer  bought  248  pounds  of  wool  of  one 
customer,  and  361  pounds  of  another ;  he  then  worked 
up  430  pounds :  how  many  pounds  had  he  left  ? 

76.  A  man  paid  375  dollars  for  a  span  of  horses,  and 
450  dollars  for  a  carriage ;  he  afterwards  sold  his  horses 
and  carriage  for  1000  dollars;  how  much  did  he  make 
by  his  bargain  ? 

77.  A  grocer  bought  285  pounds  of  lard  of  one  farmer, 
and  327   pounds  of   another;    he  afterwards  sold  110 
pounds  to  one  customer,  and  163  pounds  to  another :  how 
much  lard  did  he  have  left  ? 

78.  A  flour  dealer  having  500  barrels  of  flour  on  hand, 
sold  263  barrels  to  one  customer  and  65  barrels  to  an- 
other :  how  many  barrels  had  he  left  ? 

79.  Harriet  wished  to  read  a  book  through  which  con- 
tained 726  pages,  in  three  weeks ;  the  first  week  she  read 
165  pages,  and  the  second  week  she  read  264  pages : 
how  many  pages  were  left  for  her  to  read  the  third  week  ? 

80.  A  man  bought  a  house  for  1200  dollars,  and  hav- 
ing laid  out  210  dollars  for  repairs,  sold  it  for  1300  dol- 
lars :  how  much  did  he  lose  by  the  bargain  ? 

81.  A  young  man  having  2000  dollars,  spent  765  the 
first  year  and  843  the  second  year :  how  much  had  he 
left?' 


36  MULTIPLICATION.  [SECT.  IV, 

SECTION  IV. 

MULTIPLICATION. 

ART.  33.  Ex.  1.  What  will  three  lemons  cost,  at  2 
cents  apiece  ? 

Analysis. — Since  1  lemon  costs  2  cents,  3  lemons  will 
cost  3  times  2  cents  ;  and  3  times  2  cents  are  6  cents. 
Therefore,  3  lemons,  at  2  cents  apiece,  will  cost  6  cents. 

OBS.  The  preceding  operation  is  a  short  method  of  finding  how 
much  2  cents  will  amount  to,  when  repeated  or  taken  3  times,  and 
is  called  Multiplication.  Thus,  2  cents  -J-  2  cents  -f-  2  cents  are 
6  cents.  Hence, 

34r»  MUTIPLICATION  is  the  process  of  finding  the  amount 
of  a  number  repeated  or  added  to  itself,  a  given  number  of 
times. 

The  number  to  be  repeated  or  multiplied,  is  called  the 
multiplicand. 

The  number  by  which  we  multiply,  is  called  the  mul- 
tiplier, and  shows  how  many  times  the  multiplicand  is  to 
be  repeated  or  taken. 

The  answer,  or  number  produced  by  multiplication,  is 
called  the  product. 

Thus,  when  we  say  5  times  7  are  35,  7  is  the  multipli- 
cand, 5  the  multiplier,  and  35  the  product. 

OBS.  When  the  multiplicand  denotes  things  of  one  kind,  or  de- 
nomination only,  the  operation  is  called  Simple  Multiplication. 


QUEST. — 34.  What  is  multiplication  1  What,  is  the  numbr  r  to  be  repeated 
or  multiplied  called  1  What  the  number  by  which  we  multijyly  ?  What  does 
the  multiplier  show  ?  What  is  the  answer  called  ?  When  wo  say  5  times  7 
are  35,  which  is  the  multiplicand?  Which  the  multiplier?  Which  the 
product^  Obs.  When  the  multiplicand  denotes  things  of  one  denomination 
only,  what  is  the  operation  called  1 


ARTS.  36 — 39.]          MULTIPLICATION.  39 

38»  The  product  cf  any  two  numbers  will  be  the  same, 
whichever  factor  is  taken  for  the  multiplier.  Thus, 

If  a  garden  contains  3   rows  of  trees  as 
represented   by  the  number  of  horizontal     *  *  *  *  * 

f      JL  :        ^  '  1  -L  *     *     *     *     * 

rows  of  stars  m  the  margin,  and  each  row      . 

has  5  trees  as  represented  by  the  number  of 

stars  in  a  row,  it  is  evident,  that  the  whole 

number  of  trees  in  the  garden  is  equal  either  to  the  number 

of  stars  in  a  horizontal  row,  taken  three  times,  or  to  the 

number  of  stars  in  a  perpendicular  row  taken  five  times; 

that  is,  equal  to  5  X  3,  or  3X5. 

EXAMPLES. 

39.   When  the  multiplier  contains  but  ONE  figure. 

Ex.  1.  What  will  3  horses  cost,  at  123  dollars  apiece? 

Analysis.  —  Since  1  horse  costs  123  dollars,  3  horses 
will  cost  3  times  123  dollars. 

Directions. — Write  the  multi-  Operation, 

plicr   under    the    multiplicand;  123  multiplicand, 

then,    beginning    at    the    right  3  multiplier. 

hand,  multiply  each  figure  of  the 

u.  ,.       \\       .,    '       1f.   r       Dolls.  369  product. 
multiplicand    by  the  multiplier. 

Thus,  3  times  3  units  are  9  units,  or  we  may  simply  say 
3  times  3  a^e  9  ;  set  the  9  in  units'  place  under  the  figure 
multiplied.  3  times  2  are  6  ;  set  the  6  in  tens'  place. 
3  times  1  are  3  ;  set  the  3  in  hundreds'  place. 

Note. — The  pupil  should  be  required  to  analyze  every  example, 
and  to  give  the  reasoning  in  full ;  otherwise  the  operation  is  liable 
to  become  mer*  guess-icork,  and  a  habit  is  formed,  which  is  alike 
destructive  to  mental  discipline  and  all  substantial  improvement. 

Solve  the  following  examples  in  a  similar  manner : 
(2.)  (3.)  (4.)  (5.) 

Multiplicand     34  312  2021  1110 

Multiplier  2345 


40  MULTIPLICATION.  [SECT.  IV 

(6.)  (7.)  (8.)  (9.) 

Multiplicand,  4022  6102  7110  8101 

Multiplier,        _J3  _4  5  7 

10.  What  will  6  cows  cost  at  23  dollars  apiece. 

Suggestion. — In  this  example  the  product  of  the  differ- 
ent figures  of  the  multiplicand  into  the  multiplier,  exceeds 
9 ;  we  must  therefore  write  the  unit*'  figure  under  the 
figure  multiplied,  and  carry  the  tens  to  the  next  product 
on  the  left,  as  in  addition.    Thus,  begin- 
ning at  the  right  hand  as  before,  6  times         Operation. 
3  units  are  18  units,  or  we  may  simply  23  dolls, 

say  6  times  3  are  18.     Now  it  requires  6 

two  figures  to  express  18;  we  there-  Ans.  138  dollars* 
fore  set  the  8  under  the  figure  multi- 
plied, and  reserving  the  1,  carry  it  to  the  product  of  the 
next  figure,  as  in  addition.  (Art.  23.)  Next,  6  times  2 
are  12,  and  1  (to  carry)  makes  13.  Since  there  are  no 
more  figures  to  be  multiplied,  we  set  down  the  13  in  full. 
The  product  is  138  dollars.  Hence, 

4O»  When  the  multiplier  contains  but  one  figure. 

Write  the  multiplier  under  the  multiplicand,  units  un- 
der units,  and  draw  a  line  beneath  them. 

Begin  with  the  units,  and  multiply  each  figure  of  the 
multiplicand  by  the  multiplier,  setting  down  the  result  and 
carrying  as  in  addition.  (Art.  23.) 

Multiply  the  following  numbers  together. 

11.  78X4.  18.   524X6. 

12.  96X5.  19.  360X7. 

13.  83X3.  20.   475X4. 

14.  120X7.  21.  792X5. 

15.  138X6.  22.   820X.8. 

16.  163X5.  23.   804x7. 

17.  281X8.  24.   968X9. 


ARTS.  40,  41.]  MULTIPLICATION.  41 

25.  What  will  11 5  barrels  of  flour  cost,  at  6  dollars 
per  barrel  ? 

26.  A  man  bought  460  pair  of  boots,  at  5  dollars  a  pair : 
,10  w  much  did  he  pay  for  the  whole  ? 

27.  What  cost  196  acres  of  land,  at  7  dollars  per  acre? 

28.  What  cost  310  ploughs,  at  8  dollars  apiece? 

29.  What  cost  691  hats,  at  7  dollars  apiece? 

30.  What  cost  865  heifers,  at  9  dollars  per  head? 

31.  What  cost  968  cheeses,  at  8  dollars  apiece? 

32.  What  cost  1260  sheep,  at  7  dollars  per  head? 

33.  What  cost  9  farms,  at  2365  dollars  apiece? 

4: 1  •   When  the  multiplier  contains  more  than  ONE  figure. 

34.  A  man  sold  23  sleighs,  at  54  dollars  apiece :  how 
much  did  he  receive  for  them  all  ? 

Suggestion. — Eeasoning  as  before,    if   1   sleigh  costs 
54  dollars,  23  sleighs  will  cost  23  times  as  much. 

Directions. — As  it  is  not  Operation. 

convenient  to  multiply  by  23  54  Multiplicand, 

at  once,  we  first  multiply  by  23  Multiplier, 

the  3  units,  then  by  the  2  162  cost  of  3  s. 

tens,  and  add-the  two  results  108       "     "  20  s. 

together.     Thus,  3  times  4     Dolls.  1242     "     "23  s. 
are  12,  set  the  2  under  the 

figure  3,  by  which  we  are  multiplying,  and  carry  the  1 
as  above.  3  times  5  are  15,  and  1  (to  carry)  makes  16. 
Next,  we  multiply  by  the  2  tens  thus :  20  times  4  units 
are  80  units  or  8  tens ;  or  we  may  simply  say  2  times  4 
are  8.  Set  the  8  under  the  figure  2  by  which  we  are 
multiplying,  that  is,  in  tens'  place,  because  it  is  tens. 
2  times  5  are  10.  Finally,  adding  these  two  products 
together  as  they  stand,  units  to  units,  tens  to  tens,  &c., 
we  have  1242  dollars,  which  is  the  whole  product  re- 
quired. 


42 


MULTIPLICATION. 


[SECT.  IV 


Note. — When  She  multiplier  contains  more  than  one  figure,  the 
several  products  of  the  multiplicand  into  the  separate  figures  of  the 
multiplier,  are  called  partial  products. 

35.  Multiply  45  by  36,  and  prove  the  operation. 

Operation. 

Beginning  at  the  right  hand,  we 
proceed  thus :  6  times  5  are  30 ; 
set  the  0  under  the  figure  by  which 
we  ar£  multiplying ;  6  times  4  are 
24  and  3  (to  carry)  are  27,  &c. 


45  Multiplicand 
36  Multiplier. 
"270 
135 


PROOF. — We  multiply  the  mul- 
tiplier by  the  multiplicand,  and 
since  the  result  thus  obtained  is 
the  same  as  the  product  above, 
the  work  is  rio-ht. 


1620  Prod. 

Proof. 
36 
45 
180 
144 


1620  Prod. 

36.  What  is  the  product  of  234  multiplied  by  165  ? 

Operation. 

Suggestion. — Proceed  in  the  same  man-  234 

ner  as  when  the  multiplier  contains  but  165 

two  figures,   remembering  to  place  the         1170 
right  luand  figure  of  each  partial  product       1404 
directly  under  the  figure  by  which  you       234 
multiply.  38610  Ant 

37.  What  is  the  product  of  326  multiplied  by  205  ? 

Suggestion. — Since   multiplying   by  a  Operation. 

cipher  produces  nothing,  in  the  operation  326 

we  omit  the  0  in  the  multiplier.     Thus,  205 

having  multiplied  by  the  5  units,  we  next  ]  630 

multiply  by  the  2  hundreds,  and  place  the  652 
first  figure  of  this  partial  product  under  •    66830  Ans, 
the  figure  by  which  we  are  multiplying. 


ARTS.  42,  43.]  MULTIPLICATION.  43 

42»  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL   RULE   FOR   MULTIPLICATION 

I.  Write  the  multiplier  under  the  multiplicand,  units 
under  units,  tens  under  tens,  &c. 

II.  When  the  multiplier  contains  but  ONE  figure,  begin 
with  the  units,  and  multiply  each  figure  of  the  multipli- 
cand by  the  multiplier,  setting  down  the  result  and  carry- 
ing  as  in  addition.     (Art.  23.) 

III.  If  the  multiplier  contains  MORE  than  on,e  figure, 
multiply  each  figure  of  the  multiplicand  by  each  figure 
of  the  multiplier  separately,  and  write  the  first  figure  of 
each  partial  product  under  the  figure  by  which  you  are 
multiplying. 

Finally,  add  the  several  partial  products  together,  and 
the  sum  will  be  the  whole  product,  or  answer  required. 

43»  PROOF. — Multiply  the  multiplier  by  the  multipli- 
cand, and  if  the  second  result  is  the  same  as  the  first,  the 
work  is  right. 

OBS.  1.  It  is  immaterial  as  to  the  result  which  of  the  factors  is 
taken  for  the  multiplier.  (Art.  88.)  But  it  is  more  convenient  and 
therefore  customary  to  place  the  larger  number  for  the  multipli- 
cand and  the  smaller  for  the  multiplier.  Thus,  it  is  easier  to  mul- 
tiply 254672381  by  7,  than  it  is  to  multiply  7  by  254672381,  but 
the  product  will  be  the  same. 

2.  Multiplication  may  also  be  proved  by  division,  and  by  casting 
out  the  nines;  but  neither  of  these  methods  can  be  explained  here 
•without  anticipating  principles  belonging  to  division,  with  which 
the  learner  is  supposed  as  yet  to  be  unacquainted. 


QUEST.— 42.  How  do  you  write  numbers  for  multiplication?  When  the 
multiplier  contains  but  one  figure,  how  do  you  proceed  ?  When  the  multi- 
plier contains  more  than  one  figure,  how  proceed?  41.  JVbt«.  What  \&  meant 
by  partial  products?  What  is  to  be  done  with  the  partial  products?  43, 
How  is  multiplication  proved  ? 


44  MULTIPLICATION.  [^ECT.  IV 

EXAMPLES    FOR    PRACTICE. 

1.  Multiply  63  by  4.  10.  Multiply  46  by  10. 

2.  Multiply  78  by  5.  11.  Multiply  52  by  11. 

3.  Multiply  81  by  7.  12.  Multiply  68  by  12. 

4.  Multiply  97  by  6.  13.  Multiply  84  by  13. 

5.  Multiply  120  by  7.  14.  Multiply  78  by  15. 

6.  Multiply  231  by  5.  15.  Multiply  95  by  23. 

7.  Multiply  446  by  8.  16.  Multiply  129  by  35. 

8.  Multiply  307  by  9.  17.  Multiply  293  by  42. 

9.  Multiply  560  by  7.  18.  Multiply  461  by  55. 

19.  If  1  barrel  of  flour  costs  9   dollars,   how  much 
will  38  barrels  cost  ? 

20.  If  1  apple-tree  bears  14  bushels  of  apples,  how 
many  bushels  will  24  trees  bear  ? 

21.  In  1  foot  there  are  12  inches :  how  many  inches 
are  there  in  28  feet? 

22.  In  1  pound  there  are  20  shillings :  how  many  shil- 
lings are  there  in  31  pounds  ? 

23.  What  will  17  cows  cost,  at  23  dollars  apiece  ? 

24.  What  will  25  tons  of  hay  cost,  at  1 9  dollars  per  ton  ? 

25.  What  will  37  sleighs  cost,  at  43  dollars  apiece  ? 

26.  What  will  a  drove  of  150  sheep  come  to,  at  13 
shillings  per  head  ? 

27.  What  cost  105  acres  of  land,  at  15  dollars  per  acre? 

28.  How  much  will  135  yards  of  cloth  come  to,  at  18 
shillings  per  yard  ? 

29.  In  1  pound  there  are  16  ounces :  how  many  ounces 
are  there  in  246  pounds  ? 

30.  A  drover  sold  283  oxen,  at  38  dollars  per  head: 
how  much  did  he  receive  for  them  ? 

31.  If  you  walk  22  miles  per  day,  how  far  will  you 
walk  in  305  days  ? 

32.  In  one  day  there  are  24  hours :  how  many  hours 
are  there  in  365  days  ? 


ARTS.  44, 45.]      MULTIPLICATION.  45 

33.  In  1  year  there  are  52  weeks :  how  many  weeks 
are  there  in  175  years? 

34.  In  1  hour  there  are  60  minutes :  how  many  min- 
utes are  there  in  396  hours? 

35.  In  1  hogshead  there  are  63  gallons :  how  many 
gallons  are  there  in  450  hogsheads  ? 

36.  What  will  475  horses  cost,  at  73  dollars  apiece? 

37.  In  1  square  foot  there  are  144  square  inches :  how 
many  square  inches  are  there  in  235  feet  ? 

38.  How  far  will  a  ship  sail  in  158  days,  if  she  sails 
165  miles  per  day  ? 

44.  It  is  a  fundamental  principle  of  notation,  that 
each  removal  of  a  figure  one  place  towards  the  left,  in- 
creases its  value  ten  times;  (Art.  10;)  consequently  an- 
nexing a  cipher  to  a  number,  increases  its  value  ten  times, 
or  multiplies  it  by  10;    annexing  two  ciphers,  increases 
its  value  a  hundred  times,  or  multiplies  it  by  100 ;  an- 
nexing three  ciphers,  increases  it  a  thousand  times,  or  mul- 
tiplies it  by  1000,  <fec. ;  for  each  cipher  annexed,  removes 
each  figure  in  the  number  one  place  towards  the  left. 
Thus,    12  with  a  cipher  annexed  becomes  120,  and  i? 
the  same  as  12X10;  12  with  two  ciphers  annexed,  be- 
comes 1200,  and  is  the  same  as  12X100;  12  with  three 
ciphers  annexed,  becomes  12000,  and  is  the  same  as 
12X1000,  &c.     Hence, 

45.  To  multiply  by  10,  100,  1000,  &c. 

Annex  as  many  ciphers  to  the  multiplicand  as  there  are 
ciphers  in  the  multiplier,  and  the  number  thus  formed  will 
t>e  the  product  required. 

Note. — To  annex  means  to  place  after,  or  at  the  right  hand. 

QUEST. — 44.  What  effect  does  it  have  to  remove  a  figure  />ne  place  towards 
the  left  hand  ?  Two  piaces  ?  45.  How  do  you  proceed  when  the  multiplier 
Is  10,  100,  1000,  &c  ?  Note.  What  is  the  meaning  of  the  term  annex? 


46  MULTIPLICATION.  [SECT.  IT 

40.  What  will  10  dresses  cost,  at  18  dollars  apiece? 

Solution. — If  1  dress  costs  18  dollars,  10  dresses  will 
cost  10  times  18  dollars.  But  annexing  a  cipher  to  a 
number  multiplies  it  by  10.  We  therefore  annex  a  cipher 
to  the  multiplicand,  (18  dollars,)  and  it  becomes  180  dol- 
ars.  The  answer  therefore  is  180  dollars. 

Multiply  the  following  numbers  in  a  similar  manner : 

41.  26X10.  46.  469X10000. 

42.  37X100.          47.  523X100000. 

43.  51X1000.         48.  681X1000000. 

44.  226X1000.        49.  85612X10000. 

45.  341X1000.  50.  960305X100000. 
51.  What  will  20  wagons  cott,  at  67  dollars  apiece? 

Suggestion. — Since  multiplying  by        Operation, 
ciphers  produces  ciphers,  we  omit  mul-  67 

tiplying  by  the  0,  and  placing  the  sig-  20 

nificant  figure  2  under  the  right  hand    Ans.  1340  dollars, 
figure  of   the  multiplicand,  multiply 
by  it  in  the  usual  way,  and  annex  a  cipher  to  the  product. 
The  answer  is  1340  dollars.     Hence, 

46.  When  there  are  ciphers  on  the  right  hand  of  the 
multiplier. 

Multiply  the  multiplicand  by  the  significant  figures  of 
the  multiplier,  and  to  this  product  annex  as  many  ciphers, 
as  are  found  on  the  right  hand  of  the  multiplier. 

(52.)        (53.)        (54.)       (55.) 
85        97         123       234 
200       3000        40000      50000 


(56.) 
261 

(57.) 
329 

(58.) 
462 

(59.) 
571 

130 

2400 

3501/0 

460000 

Q.UE»T. — 46.  When  there  are  ciphers  on  the  right  of  the  multiplier,  how  da 
you  proceed  ? 


ARTS.  46,  47.]        MULTIPLICATION.  47 

60.  In  one  hour  there  are  60  minutes :  how  many  min- 
utes are  there  in  125  hours? 

61.  What  will  300  barrels  of  flour  cost  af  8  dollars  per 
barrel  ? 

62.  What  cost  400  yds.  of  cloth,  at  17  shills.  per  yd.  ? 

63.  If  the  expenses  of  1  man  are  135  dollars  per  month, 
how  much  will  be  the  expenses  of  200  men  ? 

64.  If  1500  men  can  build  a  fort  in  235  days,  how  long 
will  it  take  one  man  to  build  it  ? 

4:7.  When  there  are  ciphers  on  the  right  of  the  mul- 
tiplicand. 

Multiply  the  significant  figures  of  the  multiplicand  by 
tJie  multiplier,  and  to  tlie  product  annex  as  many  ciphers, 
as  are  found  on  the  right  of  the  multiplicand. 

65.  What  will  43  building  lots  cost,  at  3500  dollars  a 
lot? 

Having  placed  the  multiplier  under         Operation. 
the  significant  figures  of  the  multipli-  3500 

cand,  multiply  by  it  as  usual,  and  to  43 

the  product  thus  produced,  annex  two  105 

ciphers,  because  there  are  two  ciphers  140 

on  the  right  of  the  multiplicand.  Ans.  150500  dolls 

(66.)  (67.)  (68.)  (69.) 

1300  2400  21000  25000 

15  17  24  32 

70.  What  is  the  product  of  132000  multiplied  by  25  ? 

71.  What  is  the  product  of  430000  multiplied  by  34  ? 

72.  What  is  the  product  of  1520000  multiplied  by  43  ? 

73.  What  is  the  product  of  2010000  multiplied  by  52  ? 

74.  What  is  the  product  of  3004000  multiplied  by  61  ? 

Q.CBST.— 47.  When  there  are  ciphers  on  the  right  of  the  multiplinawl)  how 
do  you  proceed  ? 


48 


MULTIPLICATION. 


[SECT. 


48.  When  the  multiplier  and  multiplicand  both  have 
ciphers  on  the  right. 

Multiply  the  significant  figures  of  the  multiplicand  by 
the  significant  figures  of  the  multiplier,  and  to  this  pro- 
duct annex  as  many  ciphers,  as  are  found  on  the  right  of 
both  factors. 

75.  Multiply  16000  by  3200. 

Having  placed  the  significant  figures 
of  the  multiplier  under  those  of  the  mul- 
tiplicand, we  multiply  by  them  as  usual, 
and  to  the  product  thus  obtained,  annex 
five  ciphers,  because  there  are  five  ci- 
phers on  the  right  of  both  factors. 

Solve  the  following  examples : 
76.  2100X200. 
78.  12000X210. 
80.  38000X19000. 
82.  2800000X26000. 
84.  1000  milesX!40. 
86.  120  dollars  X  4200. 
88.  867  poundsX424. 
00.  6726  rodsX627. 
92.  25268  penceX4005. 


Operation. 

16000 
3200 

32 

48 

Ans.  51200000 


94.  376245X3164. 

96.  600400X7034. 

98.  432467X30005. 
100.  680539X80406. 
102.  Multiply  seventy -three 


77.  3400X130. 

79.  25000X2600. 

81.  500000X42000. 

83.  140  yards XI 6000. 

85.  20  dollars  X  35000, 

87.  75000  dolls.  X  365. 

89.  6830  feetX562. 

91.  7207  galls.  X  807. 

93.  36074  tons  X  4060. 

95.  703268X5346. 

97.  864325X6728. 

99.  4567832X27324. 
101.  7563057X62043. 
thousand  and  seven  by 


twenty  thousand  and  seven  hundred. 

103.  Multiply  six  hundred  thousand,  two  hundred  and 
three  by  seventy  thousand  and  seventeen. 

QUEST. — 48.  When  there  are  ciphers  on  tho  right  of  both  tl  e  inultipliel  and 
multiplicand,  how  proceed  ? 


ABTS.  48 — 50.]  DIVISIDN.  49 

SECTIt  N    V. 
DIVISION. 

ART.  49o  Ex.  1.  How  many  lead  pencils,  at  2  cents 
apiece,  can  I  buy  for  1 0  cents  ? 

Solution. — Since  2  cents  will  buy  1  pencil,  10  cents 
will  buy  as  many  pencils,  as  2  cents  are  contained  times  in 
10  cents  ;  and  2  cents  are  contained  in  10  cents,  5  times. 
I  can  therefore  buy  5  pencils. 

2.  A  father  bought  12  pears,  which  he  divided  equally 
among  his  3  children  :  how  many  pears  did  each  re- 
ceive ? 

Solution. — Reasoning  in  a  similar  manner  as  above,  it 
is  plain  that  each  child  will  receive  1  pear,  as  often  as  3 
is  contained  in  12  ;  that  is,  each  must  receive  as  many 
pears,  as  3  is  contained  times  in  12.  Now  3  is  contained  in 
12,  4  times.  Each  child  therefore  received  4  pears. 

OBS.  The  object  of  the  first  example  is  to  find  how  many  times 
one  given  number  is  contained  in  another.  The  object  of  the  second 
is  to  divide  a  given  number  into  several  equal  parts,  and  to  ascertain 
the  value  of  these  parts.  The  operation  by  which  they  are  solved 
is  precisely  the  same,  and  is  called  Division.  Hence, 

5O.  DIVISION  is  the  process  of  finding  how  many  times 
one  given  number  is  contained  in  another. 

The  number  to  be  divided,  is  called  the  dividend. 

The  number  by  which  we  divide,  is  called  the  divisor. 

The  ansiuer,  or  number  obtained  by  division,  is  called 
the  quotient,  and  shows  how  many  times  the  divisor  is 
contained  in  the  dividend. 


QUEST.— 50.  What  is  division  ?  What  is  the  number  to  be  divided,  called  ? 
The  number  by  which  we  divide  ?  What  is  the  answer  called?  What  does 
.ne  quotient  show  ? 

4 


50  DIVISIO.V.  [SECT.  V 

Note. — The  term  quotient  is  derived  from  the  Latin  word  qttotiea 
which  signifies  how  )ften,  or  how  many  times. 

51.  The  number  which  is  sometimes  left  after  division, 
in  called  the  remainder.     Thus,  when  we  say  4  is  con- 
tamed  in  21,  5  times  and  1  over,  4  is  the  divisor,  21  the 
dividend,  5  the  quotient,  and  1  the  remainder. 

OBS.  1.  The  remainder  is  always  less  than  the  divisor;  for  if  it 
were  equal  to,  or  greater  than  the  divisor,  the  divisor  could  be  con- 
tained once  more  in  the  dividend. 

2.  The  remainder  is  also  of  the  same  denomination  as  the  divi- 
dend; for  it  is  a  part  of  it. 

52.  Sign  of  Division  (-r).     The  sign  of  Division  is 
a  horizontal  line  between  two  dots  (-7-),  and  shows  that 
the   number   before   it,  is  to  be  divided  by  the  number 
after  it.     Thus,  the  expression  24—6,  signifies  that  24  is 
to  be  divided  by  6. 

Division  is  also  expressed  by  placing  the  divisor  under 
the  dividend  with  a  short  line  between  them.  Thus  the 
expression  A7*,  shows  that  35  is  to  be  divided  by  7,  and  is 
equivalent  to  35-7-7. 

53*  It  will  be  perceived  that  division  is  similar  in  prin- 
ciple to  subtraction,  and  may  be  performed  by  it.  For 
instance,  to  find  how  many  times  3  is  contained  in  12, 
subtract  3  (the  divisor)  continually  from  12  (the  dividend) 
until  the  latter  is  exhausted ;  then  counting  these  repeated 
subtractions,  we  shall  have  the  true  quotient.  Thus,  3 
from  12  leaves  9 ;  3  from  9  leaves  6 ;  3  from  6  leaves  3  ; 
3  from  8  leaves  0.  Now,  by  counting,  we  find  that  3  has 


QUEST. — 51.  What  is  the  number  called  which  is  sometimes  left  after  divi- 
sion? When  we  say  4  is  in  21,  5  times  and  1  over,  what  is  the  4  cabled?  The 
21  ?  The  5  ?  The  1 V  Obs.  Is  the  remainder  greater  or  less  than  the  divisor? 
Why?  Of  what  denomination  is  it?  Why?  52.  What  is  the  sign  of  divi- 
aion  ?  What  does  it  show  ?  In  what  other  way  is  division  expressed  ? 


ARTS.  51 — 53.]          DIVISION. 


51 


been  taken  from  12,  4  times;  consequently  3  is  contained 
t~  ?2,  4  times.     Hence, 

Division  is  sometimes  defined  to  "be  a  short  way  of  per- 
forming repeated  subtractions  of  the  same  number. 

OBS.  1.  It  will  also  be  observed  that  division  is  the  reverse  of 
multiplication.  Multiplication  is  the  repeated  addition  of  the  same 
number ;  division  is  the  repeated  subtraction  of  the  same  number. 
The  product  of  the  one  answers  to  the  dividend  of  the  other :  but 
the  latter  is  always  given,  while  the  former  is  required. 

2.  When  the  dividend  denotes  things  of  one  kind,  or  denominar 
tion  only,  the  operation  is  called  Simple  Division. 

DIVISION  TABLE. 


1  is  in 

2  is  in 

3  is  in 

4  is  in 

5  is  in 

1,  once. 

2,  once. 

3,  once. 

4,  once. 

5,  once. 

2,           2 

4,          2 

6,          2 

8,          2 

10,          2 

3,          3 

6,          3 

9,          3 

12,          3 

15,          3 

4,          4 

8,          4 

12,          4 

16,          4 

20,          4 

5,           5 

10,           5 

15,          5 

20,          5 

25,           5 

6,           6 

12,          6 

18,          6 

24,           6 

30,           6 

7,         7 

14,          7 

21,           7 

28,          7 

35,          7 

8,           8 

16,           8 

24,           8 

32,           8 

40,           8 

9,          9 

18,           9 

27,           9 

36,           9 

45,           9 

10,        10 

20,         10 

30,         10 

40,         10 

50,        10 

6  is  in 

7  is  in 

8  is  in 

9  is  in 

10  is  in 

6,  once. 

7,  once. 

8,  once. 

9,  once. 

10,  once. 

12,          2 

14,          .2 

16,          2 

18,          2 

20,        2 

18,          3 

21,          3 

24,          3 

27,          3 

30,        3 

24,          4 

28,          4 

32,          4 

36,           4 

40,        4 

30,          5 

35,          5 

40,           5 

45,          5 

50,        5 

36,          6 

42,           6 

48,           6 

54,          6 

60,        6 

42,          7 

49,          7 

56,           7 

63,          7 

70,        7 

48,           8 

56,           8 

64,           8 

72,          8 

80,        8 

54,           9 

63,           9 

72,          9 

81,           9 

90,        9 

60,        10 

70,         10 

80,        10 

90,         10 

100,      10 

QUEST.— -Obs.  When  the  dividend  denotes  things  cf  g>ne  denomination  only, 
Irhat  iR  the  operation  called  ? 


52  DIVISION.  [SECT.  V 

SHORT  DIVISION. 

ART.  54.  Ex.  1.  How  many  yards  of  cloth,  at  2  dol« 
lars  per  yard,  can  I  buy  for  246  dollars  ? 

Analysis. — Since  2  dollars  will  buy  1  yard,  246  dol- 
lars will  buy  as  many  yards,  as  2  dollars  are  contained 
times  in  246  dollars. 

Directions. — Write  the  divisor  on         Operation. 
the  left  of  the  dividend  with  a  curve     w*«>'-  ™v^n<L 
line  between  them;  then,  beginning  '  — 

at  the  left  hand,  proceed  thus:  2  is  ®uot'  l 
contained  in  2,  once.  As  the  2  in  the  dividend  denotes 
hundreds,  the  1  must  be  a  hundred ;  we  therefore  write 
it  in  hundreds'  place  under  the  figure  divided.  2  is  con- 
tained in  4,  2  times ;  and  since  the  4  denotes  tens,  the  2 
must  also  be  tens,  and  must  be  written  in  tens'  place.  2  is 
in  6,  3  times.  The  6  is  units ;  hence  the  3  must  be  units, 
and  we  write  it  in  units'  place.  The  answer  is  123  yards. 

Solve  the  following  examples  in  a  similar  manner : 

2.  Divide  42  by  2.  6.  Divide  684  by  2. 

3.  Divide  69  by  3.  Y.  Divide  4488  by  4. 

4.  Divide  488  by  4.  8.  Divide  3963  by  3. 

5.  Divide  555  by  5.  9.  Divide  6666  by  6. 

55»  When  the  divisor  is  not  contained  in  the  first 
figure  of  the  dividend,  we  must  find  how  many  times  it 
is  contained  hi  the  first  two  figures. 

10.  At  2  dollars  a  bushel,  how  much  wheat  can  be 
bought  for  124  dollars? 

Since  the  divisor  2,  is  not  contained  in       Operation. 
the  first  figure  of  the  dividend,  we  find         2)124 
how  many  times  it  is  contained  in  the  first     Ans.  62  bu, 
two  figures.    Thus  2  is  in  12,  6  times ;  set 
the  6  under  the  2.     Next,  2  is  in  4,  2  times.     The  an- 
Ewer  is  62  bushels. 


ARTS   54 — 57.]  DIVISION.  53 

11.  Divide  142  by  2.  13.  Divide  1648  by  4. 

12.  Divide  129  by  3.  14.  Divide  2877  by  7. 

56 »  After  dividing  any  figure  of  the  dividend,  if  there 
is  a  remainder,  prefix  it  mentally  to  the  next  figure  of  the 
dividend,  and  then  divide  this  number  as  before. 

Note. — To  prefix  means  to  place  before,  or  at  the  left  hand. 

15.  A  man   bought   42   peaches,  which   he   divided 
equally  among  his  3  children :  how  many  did  he  give  to 
each? 

When  we  divide  4  by  3,  there  is  1  re-       Operation, 
mainder.     This  we  prefix  mentally  to  the         3)42 
next  figure  of  the  dividend.     We  then  say,  14  Ans. 

3  is  in  12,  4  times. 

16.  Divide  56  by  4.  18.  Divide  456  by  6. 

17.  Divide  125  by  5.  19.  Divide  3648  by  8. 

57.  Having  obtained  the  first  quotient  figure,  if  the 
divisor  is  not  contained  in  any  figure  of  the  dividend,  place 
a  cipher  in  the  quotient,  and  prefix  this  figure  to  the  next 
one  of  the  dividend,  as  if  it  were  a  remainder. 

20.  If  hats  are  2  dollars  apiece,  how  many  can  be 
bought  for  216  dollars  ? 

As  the  divisor  is  not  contained  in  1,        Operation. 
the  second  figure  of  the  dividend,  we  2)216 

put  a  0  in  the  quotient,  and  prefix  the     Ans.  108  hats. 

1  to  the  6  as  directed  above.     Now  2 
is  in  16,  8  times. 

21.  Divide  2545  by  5.  23.  Divide  6402  by  6. 

22.  Divide  3604  by  4.  24.  Divide  4024  by  8. 
25.  A  man  divided  17  loaves  of  bread  equally  between 

2  poor  persons :  how  many  did  he  give  to  each  ? 
Suggestion. — Reasoning  as   before,  he   gave  each  as 

many  loaves  as  2  is  contained  times  in  17: 


54  DIVISION.  [S2CT.    V 

Thus,  2  is  contained  in  17,  8          Opemtion-. 
times  and  1  over;  that  is,  after  2)17 

giving  them  8  loaves  apiece,  there      Quot.  8-1  remainder, 
is  one  loaf  left  which  is  not  divid-          Ans.  8-J-  loaves. 
ed.     Now  2  is  not  contained  in  1 ; 

hence  the  division  must  be  represented  by  writing  the  2 
under  the  1,  thus  •£,  (Art.  52,)  which  jnust  be  annexed  to 
the  8.  The  true  quotient,  is  8-J.  He  therefore  gave  eight 
and  a  half  loaves  to  each.  Hence, 

58»  When  there  is  a  remainder  after  dividing  the  last 
figure  of  the  dividend,  it  should  always  be  written  over  the 
divisor  and  annexed  to  the  quotient. 

Note. — To  annex  means  to  place  after,  or  at  the  riglit  hand. 

59*  When  the  process  of  dividing  is  carried  on  in  the 
mind,  and  the  quotient  only  is  set  down,  the  operation  it 
called  SHORT  DIVISION. 

6O»  From  the  preceding  illustrations  and  principles,  we 
derive  the  following 

RULE   FOR   SHORT   DIVISION. 

I.  Write  the  divisor  on  the  left  of  the  dividend,  with  a 
curve  line  between  them. 

Beginning  at  the  left  hand,  divide  each  figure  of  the 
dividend  by  the  divisor,  and  place  each  quotient  figure 
under  the  figure  divided. 

II.  When  there  is  a  remainder  after  dividing  any  fig- 
ure, prefix  it  to  the  next  figure  of  the  dividend  and  divide 
this  number  as  before.     If  the  divisor  is  not  contained  in 

QUEST.— 59.  What  is  Short  Division  ?  GO.  How  do  you  write  numbers 
for  short  division?  Where  begin  to  divide  ?  Where  place  each  quotient  fig- 
ure? When  there  is  a  remainder  after  dividing  a  figure  of  the  dividend, 
what  must  be  done  with  it  1  If  the  divisor  is  not  contained  in  a  fl  ore  of  th*> 
dividend,  how  proceed?  When  there  is  a  remainder,  after  dividing  the  luat 
fl#me  of  the  dividend,  what  must  be  done  with  it  ? 


ARTS.  58 — 61.]  DIVISION.  55 

any  figure  of  the  dividend,  place  a  cipher  in  the  quotient, 
and  prefix  this  figure  to  the  next  one  of  the  dividend,  as  if 
it  were  a  remainder.  (Arts.  56,  57.) 

III.  When  there  is  a  ramainder  after  dividing  the  last 
figure y  write  it  over  the  divisor  and  annex  it  to  the  quotient, 

61»  PROOF. — Multiply  the  divisor  by  the  quotient,  to 
the  product  add  the  remainder,  and  if  the  sum  is  equal  to 
the  dividend,  the  work  is  right. 

OBS.  Division  may  also  be  proved  by  subtracting  the  remainder, 
if  any,  from  the  dividend,  then  dividing  the  result  by  the  quotient. 

EXAMPLES    FOR    PRACTICE. 

1.  Divide  426  by  3.  10.  Divide  3640  by  5. 

2.  Divide  506  by  5.  11.  Divide  6210  by  4. 

3.  Divide  304  by  4.  12.  Divide  7031  by  7. 

4.  Divide  450  by  6.  13.  Divide  2403  by  6. 

5.  Divide  720  by  7.  14.  Divide  8131  by  9. 

6.  Divide  510  by  9.  15.  Divide  7384  by  8. 

7.  Divide  604  by  5.  16.  Divide  8560  by  7. 

8.  Divide  760  by  8.  17.  Divide  7000  by  8. 

9.  Divide  813  by  7.  18.  Divide  9100  by  9. 

19.  How  many  pair  of  shoes,  at  2  dollars  a  pair,  cau 
you  buy  for  126  dollars  ? 

20.  How  many  hats,  at  4  dollars  apiece,  can  be  bought 
for  168  dollars? 

21.  A  man  bought  144  marbles  which  he  divided  equally 
among  his  6  children :  how  many  did  each  receive  ? 

22.  A  man  distributed  360  cents  to  a  company  of  poor 
children,  giving  8  cents  to  each  :  how  many  children  were 
there  in  the  company  ? 

23.  How  many  yards  of  silk,  at  6  shillings  per  yard, 
can  I  buy  for  450  shillings  ? 

QUEST.— 61.  How  is  division  proved?  Obs*  What  other  wav  of  proving 
division  is  mentioned? 


56  DIVISION.  [SECT.  V 

24.  A  man  having  600  dollars,  wished  to  lay  it  out 
in  flour,  at  7  dollars  a  barrel:  how  many  whole  barrels 
could  he  buy,  and  how  many  dollars  would  he  have  left  ? 

25.  If  you  read  9  pages  each  day,  how  long  will  it 
take  you  to  read  a  book  through  which  has  828  pages? 

26.  If  I  pay  8  dollars  a  yard  for  broadcloth,  how  many 
yards  can  I  buy  for  1265  dollars? 

27.  If  a  stage  coach  goes  at  the  rate  of  8  miles  per 
hour,  how  long  will  it  be  in  going  1560  miles  ? 

28.  If  a  ship  sails  9  miles  an  hour,  how  long  will  it 
be  in  sailing  to  Liverpool,  a  distance  of  3000  miles  ? 

LONG  DIVISION. 

ART.  62.  Ex.  1.  A  man  having  156  dollars  laid  it 

out  in  sheep  at  2  dollars  apiece :  how  many  did  he  buy  ? 

Analysis. — Reasoning  as  before,  since  2  dollars  will 

buy  1  sheep,  156  dollars  will  buy  as  many  shee$  as  2 

dollars  are  contained  times  in  156  dollars. 

Directions. — Having  written  the  di-          Operation. 
visor  on  the  left  of  the  dividend  as  in    Di™-  Divid-  ^uot- 
short  division,  proceed  in  the  follow-  34 . 

ing  manner :  — — 

First.  Find  how  many  times  the  -^ 

divisor  (2)  is  contained  in  (15)  the 
first  two  figures  of  the  dividend,  and  place  the  quotient 
figure  (7)  on  the  right  of  the  dividend  with  a  curve  line 
between  them.  Second.  Multiply  the  divisor  by  the 
quotient  figure,  (2  times  7  are  14,)  and  write  the  product 
(14)  under  the  figures  divided.  Third.  Subtract  the 
product  from  the  figures  divided.  (The  remainder  is  1.) 
Fourth.  Bringing  down  the  next  figure  of  the  dividend, 
and  placing  it  on  the  right  of  the  remainder  we  have  16. 
Now  2  is  contained  in  16,  8  times;  place  the  8  on  the 
right  hand  of  the  last  quotient  figure,  and  multiplying 


ARTS.  62,  G 3.]  DIVISION.  57 

the  divisor  by  it,  (8  times  2  are  16,)  set  the  product  undei 
the  figures  divided,  and  subtract  as  before.  Therefore  156 
dollars  will  buy  78  sheep,  at  2  dollars  apiece. 

63.  When  the  result  of  each  step  in  the  operation  is 
set  down,  the  process  of  dividing  is  called  LONG  DIVISION. 

It  is  the  same  in  principle  as  Short  Division.  The 
only  difference  between  them  is,  that  in  Long  Division 
the  result  of  each  step  in  the  operation  is  written  down, 
while  in  Short  Division  we  carry  on  the  whole  process 
in  the  mind,  simply  writing  down  the  quotient. 

Note. — To  prevent  mistakes,  it  is  advisable  to  put  a  dot  under 
each  figure  of  the  dividend,  when  it  is  brought  down. 

Solve  the  following  examples  by  Long  Division : 

2.  Divide  195  by  3.  Ans.  65. 

3.  Divide  256  by  2.  6,  Divide  2665  by  5. 

4.  Divide  1456  by  4.  V.  Divide  4392  by  6. 

5.  Divide  5477  by  3.  8.  Divide  6517  by  Y. 

OBS.  When  the  divisor  is  not  contained  in  the  first  two  figures  of 
the  dividend,  find  how  many  times  it  is  contained  in  the  first  threct 
or  the  fewest  figures  which  will  contain  it,  and  proceed  as  before. 

9.  How  many  times  is  13  contained  in  10519? 

Thus,  13  is  contained  in  105,          Operation. 

8  times;  set  the  8  in  the  quo-  13)l0519(809-t2sr  Ans. 
tient  then  multiplying  and  sub-  104 

tracting,    the    remainder    is    1.  119 

Bringing  down  the  next  figure  117 


we  have  11  to  be  divided  by  13.  2  rem. 

But  13  is  not  contained  in  11  ; 

therefore  we  put  a  cipher  in  the  quotient,  and  bring  down 

the  next  figure.  (Art.  57.)     Then  13  is  sontained  in  119, 

CUTEST.— 63.  What  is  long  division  ?    Wiiat  is  the  diflerence  between  long 
e«d  short  division  ? 


58  DIVISION.  [SECT  "V. 

9  times.  Set  the  9  in  the  quotient,  multiply  and  sub- 
tract, and  the  remainder  is  2.  Write  the  2  over  the  di- 
visor, and  annex  it  to  tho  quotient.  (Art.  58.) 

O4.  After  the  first  quotient  figure  is  obtained,  for 
each  figure  of  the  dividend  which  is  brought  down,  either 
a  significant  figure  or  a  cipher  must  be  put  in  the  quotient. 

Solve  the  following  examples  in  a  similar  manner : 

10.  Divide  15425  by  11.  Ans.  1402-ft-. 

11.  Divide  31237  by  15.  Ans.  2082-ft. 

65.  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

RULE  FOR  LONG  DIVISION. 

I.  Beginning  on  the  left  of  the  dividend,  find  liow  many 
times  the  divisor  is  contained  in  the  fewest  figures  that  will 
contain  it,  and  place  the  quotient  figure  on  the  right  of 
the  dividend  with  a  curve  line  between  them. 

II.  Multiply  the  divisor  by  this  figure  and  subtract 
the  product  from  the  figures  divided ;    to  the  right  of 
the  remainder  bring  down  the  next  figure  of  the  dividend. 
and  divide  this  number  as  before.     Proceed  in  this  man- 
ner till  all  the  figures  of  the  dividend  are  divided, 

III.  When  there  is  a  remainder  after  dividing  the  last 
figure,  write  it  over  the  divisor,  and  annex  it  to  the  quo- 
tient, as  in  short  division. 

OBS.  1.  Long  Division  is  proved  in  the  same  manner  as  Short 
Division. 

2.  When  the  divisor  contains  but  one  figure,  the  operation  by 
Short  Division  is  the  most  expeditious,  and  si  ould  therefore  be 
practiced;  but  when  the  divisor  contains  two  or  *r*  \re  figures,  it  will 
generally  be  the  most  convenient  to  divide  by  Long  Division. 

QUEST. — 65.  How  do  you  divide  in  long  division?  Where  place  the  quo- 
tient ?  Aftei  obtaining  the  first  quotient  figure,  how  proceed  ?  When  there  is 
a  remainder  after  dividing  the  L'ist  figure  of  the  dividend  what  must  be  done 
with  it?  Ols.  How  is  long  division  proved?  When  should  short  division 
be  used  ?  Wheii  long  division  ? 


ARTS.  64,  65.]  DIVISION.  59 

EXAMPLES   FOR   PRACTICE. 

1.  Divide  369  by  8.  10.  Divide  675  by  25. 

2.  Divide  435  by  9.  11.  Divide  742  by  31. 

3.  Divide  564  by  7.  12.  Divide  798  by  37. 

4.  Divide  403  by  10.  13.  Divide  834  by  42. 

5.  Divide  641  by  11.  14.  Divide  960  by  48. 

6.  Divide  576  by  12.  15.  Divide  1142  by  53. 

7.  Divide  274  by  13.  16.  Divide  2187  by  67. 

8.  Divide  449  by  14.  17.  Divide  3400  by  75. 

9.  Divide  617  by  15.  18.  Divide  4826  by  84. 

19.  How  many  caps,  at  7  shillings  apiece,  can  I  buy 
for  168  shillings? 

20.  How  many  pair  of  boots,  at  5  dollars  a  pair,  can 
be  bought  for  175  dollars  ? 

<}1.  A  man  laid  out  252  dollars  in  beef,  at  9  dollars  a 
barrel :  how  many  barrels  did  he  buy  ? 

22.  In  12  pence  there  is  1  shilling :  how  many  shillings 
are  there  in  198  pence? 

23.  In  20  shillings  there  is  1  pound  :  how  many  pounds 
are  there  in  2 1 5  shillings  ? 

24.  In  16  ounces  there  is  1  pound:  how  many  pounds 
are  there  in  268  ounces  ? 

25.  How  many  trunks,  at  15  shillings  apiece,  can  be 
bought  for  255  shillings  ? 

26.  If  27  pounds  of  flour  will  last  a  family  a  week, 
how  long  will  810  pounds  last  them? 

27.  How  many  yards  of  broadcloth,  at  23  shillings  per 
yard,  can  be  bought  for  756  shillings? 

28.  If  it  takes  18  yards  of  silk  to  make  a  dress,  how 
many  dresses  can  be  made  from  1350  yards? 

29.  How  many  sheep,  at  19  shillings  per  head,  can  be 
bought  for  1539  shillings? 

30.  A  farmer  having  1840  dollars,  laid  it  out  in  land, 
at  25  dollars  per  acre :  how  many  acres  did  he  buy? 


60  DIVISION.  [SECT.  V 

31.  A  man  wishes  to  invest  2562  dollars  in  Railroad 
stock :  how  many  shares  can  he  buy,  at  42  dollars  per 
share  ? 

32.  In  1  year  there  are  52  weeks:  how  many  years 
are  there  in  1640  weeks  ? 

33.  In  one  hogshead  there  are  63  gallons:  how  many 
hogsheads  are  there  in  3065  gallons  ? 

34.  If  a  man  can  earn  75  dollars  in  a  month,  he  wmany 
months  will  it  take  him  to  earn  3280  dollars  ? 

35.  If  a  young  man's  expenses  are  83  dollars  a  month, 
how  long  will  4265  dollars  support  him? 

36.  A  man  bought  a  drove  of  95  horses  for  4750  dol- 

o 

lars :  how  much  did  he  give  apiece  ? 

37.  If  a  man  should  spend  16  dollars  a  month,  how 
long  will  it  take  him  to  spend  172  dollars? 

38.  A  garrison  having  2790  pounds  of  meat,  wished  to 
have  it  last  them  3 1  days :  how  many  pounds  could  they 
eat  per  day  ? 

39.  How  many  times  is  54  contained  in  3241,  and  how 
many  over  ? 

40.  How  many  times  is  68  contained  in  7230,  and  how 
many  over  ? 

41.  How  many  times  is  39  contained  in  1042,  and  how 
many  over? 

42.  How  many  times  is  47  contained  in  2002,  and  how 
many  over? 

43.  What  is  the  quotient  of  1704  divided  by  56  ? 

44.  What  is  the  quotient  of  2040  divided  by  60  ? 

45.  What  is  the  quotient  of  2600  divided  by  49  ? 

46.  What  is  the  quotient  of  2847  divided  by  81  ? 

47.  Divide  1926  by  75.         51.  Divide  9423  by  105. 

48.  Divide  2230  by  85.          52.  Divide  13263  by  112, 

49.  Divide  6243  by  96.         53.  Divide  26850  by  123, 

50.  Divide  8461  by  99.         54.  Divide  48451  by  224. 


ARTS.  66,  67.]  DIVISION.  6! 

6G.  It  has  been  shown  that  annexing  a  cipher  to  a 
number,  increases  its  value  ten  times,  or  multiplies  it  by 
10,  (Art.  44.)  Reversing  this  process,  that  is,  removing 
a  cipher  from  the  right  hand  of  a  number,  will  evidently 
diminish  its  value  ten  times,  or  divide  it  by  10;  for,  each 
Sgure  in  the  number  is  thus  restored  to  its  original  place, 
and  consequently  to  its  original  value.  Thus,  annexing  a 
cipher  to  12,  it  becomes  120,  which  is  the  same  as  12  X 10. 
On  the  other  hand,  removing  the  cipher  from  120,  it  be- 
comes 12,  which  is  the  same  as  120—10. 

In  the  same  manner  it  may  be  shown,  that  removing 
two  ciphers  from  the  right  of  a  number,  divides  it  by  100; 
removing  three,  divides  it  by  1000;  removing  four,  di- 
vides It  by  10000,  &c.  Hence, 

67.  To  divide  by  10,  100,  1000,  &c. 

Cut  off  as  many  figures  from  the  right  hand  of  the  divi- 
dend as  there  are  ciphers  in  the  divisor.  The  remaining 
figures  of  the  dividend  will  be  the  quotient,  and  those  cut 
off  the  remainder. 

55.  Divide  2456  by  100. 

Since   there  are  2  ciphers  on  Operation. 

the  right  of  the  divisor,  we  cut       1JOO)24|56 
off  2  figures  on  the  right  of  the     Quot.  24  and  56  rera 
dividend.      The   quotient    is    24 
and  56  remainder,  or  24-1V<r- 

50.  Divide  1325  by  10.     Ans.  132  and  5  rem. 

57.  Divide  4620  by  100. 

58.  Divide  5633  by  1000. 

59.  Divide  8465  by  1000. 

60.  Divide  26244  by  1000. 

61.  Divide  136056  by  10000. 

QuEST.~6<5.  What  is  the  effect  of  annexing  a  cipher  to  a  number?  What  i* 
the  effect  of  removing  a  cipher  from  the  right  of  a  nr"aber  ?  67,  How  proceed 
wben  the  divisor  is  10, 100, 1000,  &e.? 


DIVISION* 


[SECT.  V- 


62.  Divide  2443667  by  100000. 

63.  Divide  23454631  by  1000000. 

68»  When  there  are  ciphers  on  the  right  hand  of  tho 
divisor. 

Cut  off  the  ciphers  from  the  divisor  ;  also  cut  of  as 
many  figures  from  the  right  of  the  dividend.  Then  divide 
the  remaining  figures  of  the  dividend  by  the  remaining  fig- 
ures of  the  divisor,  and  the  result  will  be  the  quotient. 

Finally,  annex  the  figures  cut  off  from  the  dividend  to 
the  remainder,  and  the  number  thus  formed  will  be  the  true 
remainder. 

64.  At  200  dollars  apiece,  how  many  carriages  can  be 
bought  for  4765  dollars  ? 


Having  cut  off  the  two  ciphers  on 
the  right  of  the  divisor,  and  two  fig- 
ures on  the  right  of  the  dividend,  we 
divide  the  47  by  2  in  the  usual  way. 

65.  Divide  2658  by  20. 

Ans.  132  and  18  rem 


Operation. 

2|OQ)47|65 

Ans.  23 — 165  rera. 


66.  3642  by  30. 
68.  76235  by  1400. 
70.  93600  by  2000. 
72.  23148  by  1200. 
74.  50382  by  1800. 
76.  894000  by  2500. 
78.  7450000  by  420000. 
80.  348676  —  235. 
82.  762005  —  401. 
84.  607507—1623. 
86.  4367238-7-2367. 
88.  8230732-7-3478. 
90.  93670S58-f-67213. 


67.  6493  by  200 
69.  82634  by  1600. 
71.  14245  by  3000. 
73.  42061  by  1500. 
75.  88317  by  2100. 
77.  9203010  by  3100. 
79.  9000000  by  300000. 
81.  467342  —  341. 
83.  506725  —  603. 
85.  736241  —  2764. 
87.  6203451-7-3827. 
89.  823o762-f-42316. 
91.  98765421-7-84327. 


QUEST.— 68.  When  there  are  ciphers  on  the  right  of  the  divisor,  how  pro 
ceed  ?    \Vhat  is  to  be  done  with  figures  cut  off  fro  in  the  dividend  ? 


ARTS.  68 — 73.]  FRACTIONS.  63 

SECTION    VI. 

FRACTIONS. 

7  1  •  When  a  number  or  thing,  as  an  apple  or  a  pear, 
is  divided  into  two  equal  parts,  one  of  these  parts  is  called 
one  half.  If  divided  into  three  equal  parts,  one  of  the 
parts  is  called  one  third  ;  if  divided  into  four  equal  parts, 
one  of  the  parts  is  called  one  fourth,  or  one  quarter  /  if 
into  ten,  tenths  ;  if  into  a  hundred,  hundredths,  &Q. 

When  a  number  or  thing-  is  divided  into  equal  parts,  as 
halves,  thirds,  fourths,  fifths,  &c.,  these  parts  are  called 
Fractions.  Hence, 

72.  A  FRACTION  denotes  a  part  or  parts  of  a  number 
or  thing. 

An  Integer  is  a  whole  number. 

Note. — The  term  fraction,  is  derived  from  the  Latin  fractio, 
which  signifies  the  act  of  breaking,  a  broken  part  or  piece.  Hence, 
fractions  are  sometimes  called  broken  numbers. 

73.  Fractions  are  commonly  expressed  by  two  num- 
bers, one  placed  over  the  other,  with  a  line  between  them. 
Thus,  one  half  is  written,  -J-;  one  third,  -J- ;  one  fourth,  J; 
three  fourths,  f- ;  two  fifths,  f ;  nine  tenths,  -f^-,  &c. 

The  number  below  the  line  is  called  the  denominator, 
and  shows  into  how  many  parts  the  number  or  thing  is 
divided. 


QUEST.— 71.  What  is  meant  by  one  half?  What  is  meant  by  one  third  7 
What  is  meant  by  a  fourth  ?  What  are  fourths  sometimes  called  ?  What  is 
meant  by  fifths  ?  By  sixths  ?  Eighths  ?  How  many  sevenths  mako  a  whole 
one  ?  How  many  tenths  ?  What  is  meant  by  twentieths  ?  By  hundrodths? 
72.  What  is  a  Fraction?  What  is  an  Integer?  73.  How  are  fractions  com- 
monly expressed  1  What  is  the  number  below  the  line  callud  ?  What  does  it 
show  ? 


64  FRACTIONS.  [SECT.  VL 

The  number  above  the  line  is  called  the  numerator,  and 
shows  how  many  parts  are  expressed  by  the  fraction. 
Thus  in  the  fraction  •§-,  the  denominator  .3,  shows  that  the 
number  is  divided  into  three  equal  parts  ;  the  numerator  2, 
shows  that  two  of  those  parts  are  expressed  by  the  fraction. 

The  numerator  and  denominator  taken  together,  are 
called  the  terms  of  the  fraction. 

7  4.  A  proper  fraction  is  a  fraction  whose  numerator  is 
less  than  its  denominator  ;  as,  -|-,  f  ,  -f-. 

An  improper  fraction  is  one  whose  numerator  is  equal 
to,  or  greater  than  its  denominator,  as  f,  f-. 

A  simple  fraction  is  a  fraction  which  has  but  one  nu- 
merator and  one  denominator,  and  may  be  proper,  or  im- 
proper ;  as,  f,  £ 

A  compound  fraction  is  a  fraction  of  a  fraction  ;  as,  J-  of 
i  of  I. 

A  complex  fraction  is  one  which  has  a  fraction  in  its 

2i  4     24-  -3- 

numerator,  or  denominator,  or  in  both  ;  as  —  ,  —  -,  —  f,  ~. 

5     5£  8f  f 

A  mixed  number  is  a  whole  number  and  a  fraction  writ- 
ten together  ;  as,  4f  ,  25i£. 

7G»  The  vato  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator.  Thus,  the  value  of 
•f  is  two  ;  of  %  is  one  ;  of  -J  is  one  third  ;  &c. 

Read  the  following  fractions,  and  name  the  kind  of  each  : 

1.  f;  f;  f  ;  f;  if;  if;  Y;  W;  W- 

2.  -foff;  f  offof-V;  *  of  «  of  «  of  75. 

QJ.    07       AI    3. 

3.  2i;  14f  ;  86«; 


QUEST.—  What  is  the  number  above  the  line  called?  What  does  it  show? 
What  are  the  denominator  and  numerator,  taken  together,  culled  ?  74.  What 
is  a  proper  fraction  ?  An  improper  traction  V  A  simple  fraction?  A  com- 
pound fraction  ?  A  complex  fraction  ?  A  mixed  number  ?  70.  What  is  II* 
value  of  a  fraction? 


ARTS.  74  78.]  FRACTIONS.  x  65 

To  find  a  fractional  part  of  a  given  number. 

Ex.  1.  If  a  loaf  of  bread  costs  4  cents,  what  will  half  a 
loaf  cost  ? 

Analysis. — If  a  whole  loaf  costs  4  cents,  1  half  a  loaf 
will  cost  1  half  of  4  cents  ;  and  1  half  of  4  cents  is  2  cents 
Half  a  loaf  of  bread  will  therefore  cost  2  cents. 

2.  If  a  pound  of  sugar  costs  12  cents,  what  will  1  third 
of  a  pound  cost  ? 

Analysis. — Reasoning  as  before,  if  a  whole  pound  costs 
12  cents,  1  third  of  a  pound  will  cost  1  third  of  12  cents; 
and  1  third  of  12  cents  is  4  cents.  Ans.  4  cents. 

77.  From  these  examples  the  learner  will  perceive  that 

A  half  of  a  number  is  equal  to  as  many  units,  as  2  is 
contained  times  in  that  number. 

A  third  of  a  number  is  equal  to  as  many  units  as  3  is 
contained  times  in  that  number. 

A  fourth  of  a  numbei  is  equal  to  as  many  units,  as  4  is 
contained  times  in  that  number,  &c.  Hence, 

78»  To  find  a  HALF  of  a  number,  divide  it  by  2. 
To  find  a  THIRD  of  a  number,  divide  it  by  3. 
To  find  a  FOURTH  of  a  number,  divide  it  by  4,  &c. 
Note. — For  mental  exercises  in  Fractions,  see  Mental  Arith- 
metic, Section  VII. 

3.  What  is  half  of  257  ? 

Dividing  257  by  2,  the  quotient  is  128  Operation. 

and  1  over.     Placing  the  1  over  the  2  2)257 
and  annexing  it  to  the  quotient,  we  ha\e          128i  -Ans* 
128-J-,  which  is  the  answer  required. 

4.  What  is  a  third  of  21  ?     33  ?     48  ?  78  !     151 ! 

5.  What  is  a  fourth  of  45  ?     68  ?     72  ?  81  ?     130  I 

6.  What  is  a  fifth  of  7o  ?     95  ?     135  ?  163  ? 

7.  What  is  an  eighth  of  73  ?     98  ?     104  ?     128  ? 

QCKBT.— 78.  How  do  you  find  a  half  of  a  number  ?    A  third  ?    A  fonit   t 
3 


66  FRACTIONS.  [SECT.  VI. 

8.  What  is  a  seventh  of  88  ?     133  ?     175  ?     250  ? 

9.  What  is  a  ninth  of  126  ?     163  ?     270  ?     316  ? 

7  9»  To  find  what  part  one  given  number  is  of  another. 

Make  the  number  called  the  part,  the  numerator,  and 
the  other  given  number  the  denominator.  The  fraction 
thus  formed  will  be  the  answer  required. 

1.  What  part  of  3  is  2  ?     Ans.  f. 

2.  What  part  of  4  is  1?    Is  2  ?    Is  3  ?    Is5? 

3.  What  part  of  7  is  2  ?     Is  4  ?     Is  5  ?     Is  6  ? 

4.  What  part  of  9  is  1  ?     Is  2  ?     Is  4 ?     Is  5? 

5.  5  is  what  part  of  11  ?     Of  12  ?     Of  13  ? 

6.  8  is  what  part  of  17  ?     Of  19  ?     Of  45  ? 

7.  15  is  what  part  of  38  ?     Of  57  ?     Of  85  ? 

8O«  A  part  of  a  number  being  given  to  find  the  whole. 

Multiply  the  given  part  by  the  number  of  parts  into 
which  the  whole  is  divided,  and  the  product  will  be  the 
answer  required. 

1.  27  is  1  ninth  of  what  number? 

Suggestion. — Since  27  is  1  ninth,  9  ninths,  or  the  whole, 
must  be  9  times  27;  and  27x9  —  243.  Ans. 

The  given  part  is  27,  and  the  Operation. 

number  of  parts  into  which  the  27  =  1  ninth, 

whole  is  divided,  is   9  ninths;  9= no.  parts, 

we  therefore  multiply  27  by  9.  Ans.  2 43= the  whole. 

2.  18  is  1  fifth  of  what  number? 

3.  23  is  1  fourth  of  what  number  ? 

4.  34  is  1  seventh  of  what  number  ? 

5.  45  is  1  fifteenth  of  what  number  ? 

6.  58  is  1  twelfth  of  what  number? 

7.  63  is  1  sixteenth  of  what  number  ? 

QUEST. — /9.  How  do   you   find  what  part  one  number  is  of  another  1 
80  When  a  part  of  a  number  is  given,  how  do  you  find  the  whole  ? 


ARTS.  79 — 83.]          FRACTIONS.  67 

Multiplying  a  whole  number  by  a  fraction. 

81.  We  have  seen  that  multiplying  by  a  whole  num- 
ber is  taking  the  multiplicand  as  many  times  as  there  are 
units  in  the  multiplier.  (Art.  36.)  On  the  other  hand, 

If  the  multiplier  is  only  a  part  of  a  unit,  it  is  plain  we 
must  take  only  a  part  of  the  multiplicand.  Hence, 

82»  Multiplying  by  a  fraction  is  talcing  a  certain 
PORTION  of  the  multiplicand  as  many  times  as  there  are 
like  portions  of  a  unit  in  the  multiplier.  That  is, 

Multiplying  by  -J-,  is  taking  1  half  of  the  multiplicand 
once.  Thus,  6Xi— 3. 

Multiplying  by  -£-,  is  taking  1  third  of  the  multiplicand 
once.  Thus,  6xi=2. 

Multiplying  by  f,  is  taking  1  third  of  the  multiplicand 
twice.  Thus,  6X1=4. 

Obs.  If  the  multiplier  is  a  unit  or  1,  the  product  is  equal  to  the 
multiplicand ;  if  the  multiplier  is  greater  than  a  unit,  the  product 
is  greater  than  the  multiplicand ;  (Art.  36  j)  and  if  the  multiplier  is 
less  than  a  unit,  the  product  is  less  than  the  multiplicand.  Hence, 

83.  To  multiply  a  whole  number  by  a  fraction. 
Divide  the  given  number  by  tJie  denominator,  and  mul- 
tiply the  quotient  by  the  numerator. 

Obs.  1.  The  result  will  be  the  same  if  vre first  multiply  the  given 
number  by  the  numerator,  then  divide  this  product  by  the  denomi- 
nator. 

2.  When  the  numerator  is  1,  it  is  unnecessary  to  multiply  by  it; 
for,  multiplying  by  1  does  not  alter  the  value  of  a  number.  (Art. 
82.  Obs.) 

QUEST.— 81.  What  is  meant  by  multiplying  by  a  whole  number?  82.  By  a 
fraction?  What  is  meant  by  multiplying  by  1?  By  4?  By  §?  By  |? 
Obs.  If  the  multiplier  is  a  unit  or  1,  what  is  the  product  equal  to  ?  When  the 
multiplier  is  greater  than  1,  how  is  the  product  compared  with  the  multipli- 
cand ?  When  less,  how  ?  83.  How  do  you  multfply  a  whole  number  by  a  frac- 
tion ?  Obs.  What  other  method  is  mentioned  ?  When  flie  muneiator  ie  1,  is 
tt  aece&sas  y  to  multiply  by  it  ?  Why  not  ? 


68  FRACTIONS.  [SECT.  VI, 

Ex.  1.  If  a  ton  of  coal  costs  9  dollars,  what  -will  •£  a 
ton  cost  ? 

Suggestion. — Since  a  whole  ton  costs       Ojperc&i&ri. 
9  dollars,  1  half  a  ton  will  cost  1  half  of  2)9 

9  dollars.      Now  2  is  contained  in  9,  4       Ans.  4£  dolls, 
times  and  1  over.     Place  the  1  over  the 
divisor  2,  and  annex  it  to  the  quotient.     (Art.  58.) 

2.  What  will  f  of  a  yard  of  cloth  cost,  at  36  shilling* 
per  yard  ? 

Suggestion. — First  find  what  1  third  First  Operation. 

of  a    yard  will    cost,  then    2   thirds.  3)36 

That  is,  divide  the  given  number  by  12 

the  denominator  3,  then  multiply  the  2 

quotient  by  the  numerator  2.  Ans.  24  shil. 

Or,  we  may  first  multiply  the  given  Second  Operation 

number  by  the  numerator,  then  di-  36 

vide  the  product  by  the  denomina-  2 

tor.     The  answer  is  the  same  as  be-  3)72 

fore.  Ans.  24  shil. 

3.  If  an  acre  of  land  costs  30  dollars,  what  will  •£•  of 
an  acre  cost  ? 

4.  What  will  -J-  of  a  barrel  of  flour  cost,  at  40  shillings 
per  barrel  ? 

5.  What  will  £  of  a  hogshead  of  molasses  cost,  at  37 
dollars  per  hogshead  ? 

6.  What  will  -f-  of  a  barrel  of  apples  cost,  at  28  shil- 
fings  per  barrel  ? 

7..  Multiply  48  by  f .  12.  Multiply  56  by  f. 

8.  Multiply  35  by  f.  13.  Multiply  72  by  f. 

9.  Multiply  54  by  i.  14.  Multiply  120  by  f. 

10.  Multiply  49  by  f.  15.  Multiply  168  by  f. 

11.  Multiply  64  by  -f.  16,  Multiply  243  by  \. 


ABT  84.]  FRACTIONS.  69 

Multiplying  a  whole  number  by  a  mixed  number. 

17.  What  will  5£  yards  of  cloth  cost,  at  18  shillings 
per  yard  ? 

Suggestion. — Since  1  yard  costs        2)18  cost  of  1  yd. 
18  shillings,  5-£  yards  will  cost  5-£  Si- 

times  as  much.     We  first  multiply  90  cost  of  5  yds. 

18  shillings  by  5,  then  by  £,  and  9     "of -J-  yd. 

add  the  products  together.    Hence,  Ans.  99s.  "  of  5|  yds. 
84.  To  multiply  a  whole  number  by  a  mixed  number 
Multiply  first  by  the  whole  number,  then  by  the  fraction, 
and  add  the  products  together.  (Art.  83.) 

18.  Multiply  26  by  2£.  Ans.  65. 

19.  Multiply  30  by  2-J-.  25.  Multiply  75  by  2-f. 

20.  Multiply  36  by  3-J-.  26.  Multiply  63  by  4-f. 

21.  Multiply  45  by  4£.  27.  Multiply  100  by  5f. 

22.  Multiply  42  by  5«K  28.  Multiply  165  by  7f. 

23.  Multiply  36  by  3*-  29.  Multiply  180  by  8f. 

24.  Multiply  56  by  H.  30.  Multiply  192  by  9$. 

31.  Multiply  41  rods  by  5-J-.  Ans.  225£  rods. 

32.  Multiply  68  rods  by  16-J-. 

33.  What  cost  21-£  acres  of  land,  at  35  dollars  pei 
acre  ? 

34.  What  cost  34-J-  hundred  weight  of  indigo,  at  47 
dollars  per  hundred  ? 

35.  What  cost  63f  tons  of  iron,  at  96  dollars  per  ton? 

Dividing  a  whole  number  by  a  fraction. 

Ex.  1.  How  many  apples  at  -J-  a  cent  apiece,  can  you 
buy  for  5  cents  ? 

Analysis. — If  %  a  cent  will  buy  1  apple,  5  cents  will 
buy  as  many  apples,  as  -J-  a  cent  is  contained  times  in  5 
cents ;  that  is,  as  many  as  there  are  halves  in  5  whole  ones, 

QUEST.— 84.  How  do  you  multiply  a  whole  number  by  a  mixed  number  ? 


70  FRACTIONS.  [SECT.  VI 

Now  in  1  cent  there  are  2  halves,  therefore  in  5  cents 
there  are  5  times  2,  which  are  10  halves  ;  and  1  half  is 
containad  in  10  halves,  10  times.  Ans.  10  apples. 

2.  How  many  plums,  at  -f  of  a  cent  apiece,  can  you 
buy  for  8  cents  ? 

Analysis.  —  Reasoning  as  before,  you  can  buy  as  many 
plums  as  |  of  a  cent  are  contained  times  in  8  cents.  Now 
in  1  cent  there  are  3  thirds,  therefore  in  8  cents  there  are 
8  times  3,  which  are  24  thirds,  and  2  thirds  are  contained 
in  24  thirds,  12  times.  Ans.  12  plums.  Hence, 

8  5  •  To  divide  a  whole  number  by  a  fraction. 

Multiply  the  given  number  by  the  denominator,  and 
divide  the  result  by  the  numerator. 

OBS.  When  the  numerator  is  1,  it  is  unnecessary  to  divide  bj 
it  ;  for  it  is  plain  that  dividing  by  1  does  not  alter  the  value  of  e 
number. 

3.  Divide  17  by  i. 

We  multiply  the  17  by  the  denominator  Operation. 
2;  and  since  dividing  by  1  does  not  alter          17 
the  value  of  a  number,  we  do  not  divide  2 

by  it.  34  Ans 

4.  Divide  19  by  -f. 

Operation. 

Multiply  the   19  by  3,  and  divide  the          19 
product   by    2.     Place   the   remainder   1  3 

over  the  divisor,  and  annex  it  to  the  28.  2)57 

" 


6.  Divide  25  by  -J-.  10.  Divide  89  by  f. 

6.  Divide  38  by  i.  11.  Divide  123  by 

7.  Divide  47  by  \.  12.  Divide  156  by 

8.  Divide  63  by  -2V  13.  Divide  190  by  - 

9.  Divide  72  by  f.  14.  Divide  256  by  - 

QUEST.  —  85.  II  ow  do  you  divide  a  whole  number  by  a  fraction  ?  (Ift*. 
the  numerator  is  1,  is  it  necessary  to  divide  by  it?    Why  not? 


ARTS.  85,  8G.]  FRACTIONS.  Yl 

Dividing  a  whole  number  by  a  mixed  number. 

Ex.  1.  How  many  lemons,  at  5-J-  cents  apiece,  can  you 
buy  for  22  cents? 

Analysis.  —  Since  5j  cents  will  buy  1  lemon,  22  cents 
will  buy  as  many  lemons,  as  5-J-  cents  are  contained  times 
in  22  cents.  Now  in  5|-  cents  there  are  11  halves,  and  in 
22  cents  there  are  44  halves  ;  but  11  halves  are  contained 
in  44  halves,  4  times.  Ans.  4  lemons. 

Suggestion.—  We    change    the    divi-     Operation,. 
sor  to  halves  by  multiplying  the  whole        51^22 
number    by   the    denominator    2,    and        2       2 
adding    the    numerator,    we   have    11      ji  ^44 
halves  ;   then  reducing  the  dividend  to     ^ns  —^  jemong 
halves  by  multiplying  it  by  2,  we  have 
44  halves.     Now  11  is  contained  in  44,  4  times.     Hence, 

86.  To  divide  a  whole  number  by  a  mixed  number. 

Multiply  the  whole  number  in  the  divisor  by  the  denomi- 
nator, and  to  the  product  add  the  numerator.  Then  mul- 
tiply the  dividend  by  the  same  denominator,  and  divide  as 
in  ivhole  numbers. 

Note.  —  For  further  illustrations  of  the  principles  of  fractions  see 
Practical  Arithmetic,  Section  VI.  It  is  incompatible  with  the 
design  of  the  present  work  to  treat  of  fractions  more  exten 
sively  than  is  necessary  to  enable  the  pupil  to  understand  the 
operations  in  Reduction. 

EXAMPLES. 

2.  How  many  times  is  4f-  contained  in  15  ? 

Suggestion.—  Multiplying  the  4  and  15      Qperationt 
by  3,  reduces  them  to  thirds.     Now  it  is 


. 

plain  we  can  divide  thirds   by  thirds  as  33 

well  as  we  can  divide  one  whole  number 


by  another  ;  for  the  divisor  is  of  the  same     _.» "^     ~Q~T 
name  or  denomination  as  the  dividend. 

QUEST. — 86.  How  "lo  you  divide  a  whole  number  by  a  mixed  number  7 


72  FRACTIONS.  [SECT.  VI 

3.  Divide  18  by  1J.  7.  Divide  46  by  7f. 

4.  Divide  20  by  3-J.  8.  Divide  60  by  5i. 

5.  Divide  25  by  5f.  9.  Divide  75  by  8f . 

6.  Divide  37  by  6-J-.  10.  Divide  100  by  lOf. 

EXAMPLES    FOB   PRACTICE. 

1.  How  many  apples  can  you  buy  for  4  cents,  if  you 
pay  •£  a  cent  apiece  ? 

2.  How  many  peaches  can  you  buy  for  6  cents,  if  you 
pay  -J  of  a  cent  apiece  ? 

3.  How  many  yards  of  tape  can  Sarah  buy  for  8  cents, 
if  she  pays  -J-  of  a  cent  a  yard  ? 

4.  How  many  yards  of  ribbon  can  Harriet  buy  for  9 
shillings,  at  -f  of  a  shilling  per  yard  ? 

5.  How  many  pounds  of  tea,  at  f  of  a  dollar  a  pound, 
can  be  bought  for  6  dollars  ? 

6.  How  many  yards  of  calico,  at  -J  of  a  dollar  per  yaid, 
can  you  buy  for  3  dollars  ? 

7.  At  •£  of  a  penny  apiece,  how  many  marbles  can 
George  buy  for  14  pence  ? 

8.  At  -f  of  a  dollar  a  bushel,  how  many  bushels  of  corn 
can  a  man  buy  for  6  dollars  ? 

9.  At  -f  of  a  dollar  a  yard,  how  many  yards  of  silk  can 
a  lady  buy  for  1 5  dollars  ? 

10.  At  -fc  of  a  dollar  apiece,  how  many  lambs  can  a 
drover  buy  for  27  dollars? 

11.  In  1  rod  there  are  5£  yards :  how  many  rods  are 
there  in  88  yards  ? 

12.  In  1  rod  there  are  !<>£  feet:  how  many  rods  are 
therein  132  feet? 

13.  How  many  yards  of  cloth,  at  5f  dollars  per  yard, 
can  be  bought  for  100  dollars? 

14.  How  many  cows,  at  12£  dollars  apiece,  can  be 
bought  for  125  dollars  ? 


ART.  86. J  FRACTIONS.  73 

15.  How  many  acres  of  land,  at  20-f-  dollars  per  acre, 
can  a  man  buy  for  540  dollars  ? 

16.  A  grocer  bought  a  quantity  of  flour  for  239  dol- 
1  irs,  which  was  8-£  dollars  per  barrel :  how  many  barrels 
did  he  buy  ? 

17.  A  merchant  bought  a  quantity  of  broadcloth,  at 
7f  dollars  per  yard,  and  paid  372  dollars  for  it:  how 
many  yards  did  he  buy  ? 

18.  A  man  hired  a  horse  and  chaise  to  take  a  ride,  and 
paid  275  cents  for  the  use  of  it,  which  was  12-J-  cents  per 
mile :  how  many  miles  did  he  ride  ? 

19.  If  a  man  hires  a  horse  and  carriage  to  go  1  Si- 
miles, and  pays  315  cents  for  it,  how  many  cents  does  he 
pay  per  mile  ? 

20.  A  young  man  hired  himself  out  for  16f  dollars  per 
month,  and  at  the  end  of  his  time  he  received  201  dollars : 
how  many  months  did  he  work  ? 

21.  A  farmer  having  261  dollars,  wished  to  lay  it  out 
in  young  cattle  which  were  worth  10-f  dollars  per  head : 
how  many  could  he  buy  ? 

22.  A  man  having  100  acres  of  land,  wishes  to  find 
how  many  building  lots  it  will  make,  allowing  -fa  of  an 
acre  to  a  lot :  how  many  lots  will  it  make  ? 

23.  How  many  barrels  of  beef,  at  9-J-  dollars  per  barrel, 
can  be  bought  for  156  dollars  ? 

24.  How  many  hours  will  it  take  a  man  to  travel  250 
miles,  if  he  goes  12-J-  miles  per  hour  ? 

25.  In  1  barrel  there  are  31£  gallons:  how  many  bar- 
rels are  there  in  315  gallons  ? 

26.  A  farmer  paid  843  dollars  for  some  colts,  which 
was  35^  dollars  apiece :  how  many  did  he  buy? 

27.  A  wagon  maker  sold  a  lot  of  wagons  for  1452  dol- 
lars, which  was  45f  dollars  apiece :   how  many  did  he 
sell? 


74  COMPOUND  [SECT.  VIL 

0 

.    SECTION  VII. 

COMPOUND   NUMBERS. 

ART.  87 •  SIMPLE  Numbers  are  those  which  express 
units  of  the  same  kind  or  denomination ;  as,  one,  two, 
three  ;  4  pears,  5  feet,  &c. 

COMPOUND  Numbers  are  those  which  express  units 
of  different  kinds  or  denominations ;  as  the  divisions  of 
money,  weight,  and  measure.  Thus,  6  shillings  and  7 
pence ;  3  feet  and  7  inches,  &c.,  are  compound  numbers. 

Note. — Compound  Numbers  are  sometimes  called  Denominate 
Numbers. 

FEDERAL  MONEY. 

88.  Federal  Money  is  the  currency  of  the  United 
States.  Its  denominations  are,  Eagles,  dollars,  dimes, 
cents,  and  mills. 

10  mills  (m.)  make  1  cent,     marked  ct. 

10  cents  "      1  dime,         "       d. 

10  dimes  "      1  dollar,        "       doll  or  $. 

10  dollars  "     1  eagle,         "       E. 

89»  The  national  coins  of  the  United  States  are  of 
three  kinds,  viz  :  gold,  silver,  and  copper. 

1.  The  gold  coins  are  the  eagle,  half  eagle,  and  quarter 
eagle,  the  double  eagle*  and  gold  dollar.* 

2.  The  silver  coins  are  the  dollar,  half  dollar,  quarter 
dollar,  the  dime,  half  dime,  and  three-cent-piece. 

QUEST. — 87.  What  are  simple  numbers  ?  What  are  compound  numbers'? 
88.  What  is  Federal  Money  ?  Recite  the  Table.  89.  Of  how  many  kinds  are 
the  coins  of  the  United  States  ?  What  are  the  gold  coins  ?  What  are  the 
silver  coins  ? 

*  Added  by  Act  of  Congress,  Feb.  20th,  1849. 


ARTS.  87 — 91.]  NUMBERS.  75 

3.  The  copper  coins  are  the  cent  and  half  cent, 
Mills  are  not  coined. 

Obs.  Federal  money  was  established  by  Congress,  August  8th, 
1786.  Previous  to  this,  English  or  Sterling  money  was  the  princi- 
pal currency  of  the  country. 

STERLING  MONEY. 

90,  English  or  Sterling  Money  is  the  national  cur- 
rency of  Great  Britain. 

4  farthings  (qr.  or  far.)  make  1  penny,  marked         d. 
12  pence  "      1  shilling,     "  s. 

20  shillings  "      1  pound  or  sovereign,  £. 

21  shillings  "      1  guinea. 

OBS.  The  Pound  Sterling  is  represented  by  a  gold  coin,  called 
a  Sovereign.  Its  legal  value,  according  to  Act  of  Congress,  1842,  is 
$4.84;  its  intrinsic  value,  according  to  assays  at  the  U.  S.  mint,  is 
$4.861.  The  legal  value  of  an  English  shilling  is  24-1  cents. 

TROY  WEIGHT. 

91.  Troy   Weight  is  used  in  weighing  gold,  silver, 
jewels,  liquors,  &c.,  and  is  generally  adopted  in  philo- 
sophical experiments. 

24  grains  (gr.)       make  1  pennyweight,  marked  pwt. 
20  pennyweights       "     1  ounce,  "       oz. 

12  ounces  "     1  pound,  "       Ib. 

Note. — Most  children  have  very  erroneous  or  indistinct  ideas  of 
the  weights  and  measures  in  common  use.  It  is,  therefore,  strongly 
recommended  for  teachers  to  illustrate  them  practically,  by  referring 
to  some  visible  object  of  equal  magnitude,  or  by 'exhibiting  the  ounce, 
the  pound ;  the  linear  inch,  foot,  yard,  and  rod ;  also  a  square  and 
cubic  inch,  foot,  &c. 

QUEST.— What  are  the  copper  coins  ?  Obs.  When  and  by  whom  was  Federal 
Money  established  ?  90.  What  is  Sterling  Money  ?  Repeat  the  Table.  Obs.  By 
what  h  the  Pound  Sterling  represented  ?  What  is  its  legal  value  in  dollars  and 
tents  ?  What  is  the  value  of  an  English  shilling  ?  91.  in  what  is  Troy  Weight 
deed  ?  Recite  the  Ttible, 


*o  COMPOUND  [SECT.  VII, 

AVOIRDUPOIS  WEIGHT. 

92.  Avoirdupois  Weiylit  is  used  in  weighing  groceries 
and  all  coarse  articles  ;  as  sugar,  tea,  coffee,  butter,  cheese, 
flour,  hay,  &c.,  and  all  metals  except  gold  and  silver. 

16  drams  (dr.)  make  1  ounce,  marked     oz. 

16  ounces  "     1  pound,       "  Ib. 

25  pounds  "     1  quarter,    "  qr. 

4  quarters,  or  100  Ibs.    "     1  hundred  weight,  ctvt. 
20  hund.,  or  2000  Ibs.      "     1  ton,  marked          T. 
OBS.  1.  Gross  weight  is  the  weight  of  goods  with  the  boxes,  or  bags 
which  contain  them,  allows  112  Ibs.  for  a  hundred  weight. 
Net  weight  is  the  weight  of  the  goods  only. 
2.  Formerly  it  was  the  custom  to  allow  112  pounds  fora  hundred 
weight,  and  28  pounds  for  a  quarter :  but  this  practice  has  become 
nearly  or  quite  obsolete.     The  laws  of  most  of  the  states,  as  well  as 
general  usage,  call  100  Ibs.  a  hundred  weight,  and  25  Ibs.  a  quarter. 
In  estimating  duties,  and  weighing  a  few  coarse  articles,  as  iron, 
dye-woods,  and  coal  at  the  mines,  112  Ibs.  are  still  allowed  for  a 
hundred  weight.     Coal,  however,  is  sold  in  cities,  at  100  Ibs.  for  a 
hundred  weight. 

APOTHECARIES'  WEIGHT. 

93.  Apothecaries'  Weight  is  used  by  apothecaries  and 
physicians  in  mixing  medicines. 

20  grains  (yr.)  make  1  scruple,  marked  sc.  or  S. 
3  scruples  "  1  dram,  "  dr.  or  3. 
8  drams  "  1  ounce,  "  oz.  or  g. 

12  ounces  "      1  pound,        "  Ib. 

OBS.  1.  The  pound  and  ounce  in  this  weight  are  the  same  as  the 
Troy  pound  and  ounce;  the  subdivisions  of  the  ounce  are  different. 
2.  Drugs   and  medicines  are  bought  and  sold  by  avoirdupois 
weight. 

QUEST.— 92.  In  what  is  Avoirdupois  Weight  used  ?  Recite  the  Table.  Obs 
What  is  gross  weight?  What  is  net  weight?  How  many  pounds  were  for- 
merly allowed  for  a  quarter  ?  How  many  for  a  hundred  weight  ?  93.  In  what 
is  Apothecaries  Weight  used?  Repeat  the  Table.  Obs.  To  what  are  the  Apo- 
thecaries' porn<l  and  o»«*ce  equal?  How  are  drugs  and  medicines  bought 
and  sold  ? 


ARTS.  92 — 95.]  NUMBERS.  77 

LONG  MEASURE. 

O4r«  Long  Measure  is  used  in  measuring  length  or 
distances  only,  without  regard  to  breadth  or  depth- 

12  inches  (in.)               make  1  foot,  marked  ft. 

3  feet                               "  1  yard,  "       yd. 

5i  yards,  or  16£  feet        "  1  rod,  perch,  or  pole,  r.  orp 

40  rods                              "  1  furlong,  marked  fur. 

8  furlongs,  or  320  rods    "  1  mile,  "      m. 

3  miles                             "  1  league,  "       L 
60  geographical  miles,  or ) 

691  statute .miles  \"    l  d^ree-          "    **'"*• 

360  deg.  make  a  great  circle,  or  the  circum.  of  the  eart li, 

OBS.  1.  4 inches  make  a  hand;  9  inches,  1  span;  18  inches,  1 
cubit ;  6  feet,  1  fathom ;  4  rods,  1  chain ;  26  links,  1  rod. 

2.  Long  measure  is  frequently  called  linear  or  lineal  measure. 
"Formerly  the  inch  was  divided  into  3  barleycorns  ;  but  the  barley- 
corn, as  a  measure,  has  become  obsolete.  The  inch  is  commonly 
divided  either  into  eighths,  or  tenths ;  sometimes  it  is  divided  into 
twelfths,  which  are  called  lines. 

CLOTH  MEASURE. 

95»  Cloth  Measure  is  used  in  measuring  cloth,  lace,  and 
all  kinds  of  goods,  which  are  bought  or  sold  by  the  yard. 

2  J  inches  (in.)  make  1  nail,  marked  na. 

4  nails,  or  9  in.     "     1  quarter  of  a  yard,      "      qr. 

4  quarters  "     1  yard,  "      yd. 

3  quarters  "     1  Flemish  ell,  "      Fl.  e. 

5  quarters  "     1  English  ell,  "      E.  e. 

6  quarters  "     1  French  ell,  "      F.  e. 


QUEST.— 94.  In  what  is  Long  Measure  used  1  Repeat  the  Table.  Draw  a 
line  an  inch  long  upon  your  slate  or  black-board.  Draw  one  two  inches  long. 
Draw  another  a  foot  long.  Draw  one  a  yard  long.  How  long  is  your  teacher7! 
desk  1  How  long  is  the  school-room  ?  How  wide  ?  Obs.  What  is  Long  Meas- 
ure frequently  called  7  How  is  the  inch  commonly  divided  at  the  present 
dav  ?  95.  In  what  is  Cloth  Measure  used  1  Repeat  the  Table. 


COMPOUND 


[SECT.  VIL 


OBS.  Cloth  mear  ire  is  a  species  of  long  measure.  The  yard  is 
the  eame  in  both.  Cloths,  laces,  <fec.,  are  bought  and  sold  by  the 
linear  yard,  without  regard  to  their  width. 

SQUARE  MEASURE. 

96*  Square  Measure  is  used  in  measuring  surfaces, 
or  things  whose  length  and  breadth  are  considered  with- 
out regard  to  height  or  depth  ;  as  land,  flooring,  plaster- 
ing, &c. 
144  square  in.  (sq.  in.)  make  1  square  foot,  marked  sq.ft. 


1  square  yard, 


sq.  yd. 


1  sq.  rod,  perch,  (( 

or  pole, 

sq. 

1  rood,                 " 

R. 

1  acre, 

A. 

1  square  mile,      " 

M. 

9  square  feet 

30-J-  square  yards,  or  )     ((    ( 
2  7  2i  square  feet          )          ( 
40  square  rods 

4  roods,  or  160  sq.rds.  " 
640  acres  " 

OBS.  1.  A  square  is  a  figure,  which  has  four  equal  sides,  and  all 
its  angles  right  angles,  as  seen  in  the  adjoining  diagram.   Hence, 

2.  A  Square  Inch  is  a  square,  whose  sides         9  sq.ft.—  I  sq.  yd. 
are  each  a  linear  inch  in  length. 

A  Square  Foot  is  a  square,  wiiose  sides 
are  each  a  linear  foot  in  length. 

A  Square  Yard  is  a  square,  whose  sides 
are  each  a  linear  yard  or  three  linear  feet  in 
length,  and  contains  9  square  feet,  as  re- 
presented in  the  adjacent  figure. 

3.  In  measuring  land,  surveyors  use  a 
chain  which  is  4  rods  long,  and  is  divided  • 

into  100  links.     Hence,  25  links  make  1  rod,  and  7-j^j-  inched 
make  1  link. 

This  chain  is  commonly  called  Counter's  Chain,  fiom  the  name 
of  its  inventor. 

4.  Square  Measure  is  sometimes  called  Land  Measure,  because 
?t  is  used  in  measuring  land. 

QUEST.—  Obs.  Of  what  is  Cloth  Measure  a  species?  96.  In  what  is  Squaro 
Measure  used  1  Repeat  the  Table.  Obs.  What  is  a  square  ?  What  is  a  square 
inch?  What  is  a  square  foot?  A  square  jard?  Can  you  draw  a 
inch  ?  Can  you  draw  a  square  foot  ?  A  square  yard  ? 


ARTS.  96,  97.] 


NUMBERS. 


79 


1  cubic  yard,        " 

cu.  yd. 

1  ton, 

T. 

1  ton  of  shipping,  " 
1  cord  foot,  or  a  M 
foot  of  wood, 

T. 
c.ft. 

1  cord, 

0. 

CUBIC  MEASURE. 

97.  Cubic  Measure  is  used  in  measuring  solid  bodies, 
or   things   which   have    length,    breadth,   and    thickness; 
such  as  timber,  stone,  boxes  of  goods,  the  capacity  of 
rooms,  &c. 
1 72  8  cubic  inches  (cu.  in.)  make  1  cubic  foot,    marked  cu.ft. 

27  cubic  feet 

40  feet  of  round,  or 

50  ft.  of  hewn  timber, 

42  cubic  feet 

16  cubic  feet 

8  cord  feet,  or  '. 
128  cubic  feet 

OBS.  1.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high, 
contains  1  cord.  For  8  into  4  into  4=128.  27  cu.  ft.=l  cu.  yd. 

2.  A  Cube  is  a  solid  body  bounded  by 
six  equal  squares.     It  is  often  called  a  hex- 
acdron.     Hence, 

A  _Cubic  Inch  is  a  cube,  each  of  whose 
sides  is  a  square  inch,  as  represented  by 
the  adjoining  figure. 

A  Cubic  Foot  is  a  cube,  each  of  whose 
sides  is  a  square  foot. 

3.  The  Cubic  Ton  is  chiefly  used  for  estimating  the  cartage  and 
transportation  of  timber.     By  a  ton  of  round  timber  is  meant,  such 
a  quantity  of  timber  in  its  rough  or  natural  state,  as  when  hewn, 
will  make  40  cubic  feet,  and  is  supposed  to  be  equal  in  weight  to 
50  feet  of  hewn  timber. 

4.  The  cubic  ton  or  load,  is  by  no  means  an  accurate  or  uniform 
standard  of  estimating  weight ;  for,  different  kinds  of  timber,  are  of 
very  different  degrees  of  density.     But  it  is  perhaps  sufficiently  ac- 
curate for  the  purposes  to  which  it  is  applied. 

QUEST.— 97.  In  what  is  Cubic  Measure  used?  Recite  the  Table.  How 
long,  wide,  and  high,  must  a  pile  of  wood  be  to  make  a  cord  ?  What  is  a 
cube  ?  What  is  a  cubic  inch  ?  What  is  a  cubic  foot  ?  Can  you  draw  a  cubio 
Inch  on  your  slate  ? 


80  COMPOUND  [SECT.  VIL 

WINE  MEASURE. 

98.  Wine  Measure  is  used  in  measuring  wine,  alco- 
hol, molasses,  oil,  and  all  other  liquids  except  beer,  ale, 
and  milk. 

4  gills  (gi.)  make  1  pint,         marked         pt. 

2  pints  "     1  quart,  "  qt. 

4  quarts  "     1  gallon,  "  gal. 

3 1£  gallons  "     1  barrel,  "  bar.orbbl. 

42  gallons  "     1  tierce,  "  tier. 

63  gallons,  or  2  bbls.  "     1  hogshead,      "  hkd. 

2  hogsheads  "     1  pipe  or  butt,  "  pi. 

2  pipes  "     1  tun,  "  tun, 

OBS    The  wine  gallon  contains  231  cubic  inches. 
BEER  MEASURE. 

99.  Beer  Measure  is  used  in  measuring  beer,  ale,  and 
milk. 

2  pints  (pt.)         make  1  quart,     marked  qt. 

4  quarts  "     1  gallon,         "  gal. 

36  gallons  "     1  barrel,         "     bar.  or  bbl. 

54  gals,  or  1^  bbls.  "     1  hogshead,    "  hhd. 

OBS.  The  beer  gallon  contains  282  cubic  inches.    In  many  place* 

milk  is  measured  by  wine  measure. 

DRY  MEASURE. 

1  GO.  Dry  Measure  is  used  in  measuring  grain,  fruit 
salt,  &c. 

2  pints  (pts.)        make  1  quart,  marked      qt. 

8  quarts  "     1  peck,  "  pJc. 

4  pecks,  or  32  qts.  "     1  bushel,  "  bu. 

8  bushels  "     1  quarter,  "  qr. 

32  bushels  "     1  chaldron,  "  cA. 

e.-t-ln  England,  36  bushels  of  coal  make  a  chaldron. 


QUEST.— 98.  In  what  is  Wine  Measure  used?  Recite  the  Table.  Obs.  Ho* 
many  cubic  inches  in  a  wine  gallon?  99.  In  what  is  Beer  Measure  uced 
Repeat  the  Table.  Obs.  How  many  cubic  inches  in  a  boer  gallon  ? 


ARTS.  98 — 102.]  NUMBERS.  81 

TIME. 

1O1  •  Time  is  naturally  divided  into  days  and  years  ; 
the  former  are  caused  by  the  revolution  of  the  Earth  on  its 
axis,  the  latter  by  its  revolution  round  the  Sun. 
60  seconds  (sec.)  make  1  minute,      marked  min. 

60  minutes  "      1  hour,  "       hr. 

24  hours  "      1  day,  "       d. 

7  days  "      1  week,  "       wk. 

4  weeks  "      1  lunar  month,  "       mo. 

12  calendar  months,  or     >    ..  .  ., 

'       x  >    "      1  civil  year,       "       yr. 
365  clays,  6  hrs.,  (nearly,)  $ 

13  lunar  mo.,  or  52  weeks,    "      1  year,  "       yr. 
100  years                                 "      1  century,          "       cen. 

OBS.  1.  Time  is  measured  by  clocks,  watches,  chronometers,  dials, 
hour-glasses,  &c. 

2.  A  civil  year  is  a  legal  or  common  year  ;  a  period  of  time  es- 
tablished by  government  for  civil  or  common  purposes. 

3.  A  solar  year  is  the  time  in  which  the  earth  revolves  round 
the  sun,  and  contains  365  days,  5  hours,  48  min.,  and  48  sec. 

4.  A  leap  year,  sometimes  called  bissextile,  contains  866  days, 
and  occurs  once  in  four  years. 

It  is  caused  by  the  excess  of  6  hours,  which  the  civil  year  con- 
tains above  365  days,  and  is  so  called  because  it  leaps  or  rims  over 
one  day  more  than  a  common  year.  The  odd  day  is  added  to  Feb- 
ruary, because  it  is  the  shortest  month.  Every  leap  year,  there- 
fore, February  has  29  days. 

1Q2.  The  names  of  the  days  are  derived  from  the 
names  of  certain  Saxon  deities,  or  objects  of  worship.  Thus, 

Sunday  is  named  from  the  sun,  because  this  day  was  dedicated 
to  its  worship. 

Monday  is  named  from  the  moon,  to  which  it  was  dedicated. 

QUEST.— 100.  In  what  is  Dry  Measure  used  ?  Recite  the  table.  101.  How 
is  Time  naturally  divided  ?  How  are  the  former  caused  ?  How  the  latter  t 
Repeat  the  Table.  Obs.  How  is  Time  measured  1  What  is  a  civil  year  ?  A 
solar  year?  A  leap  year?  How  is  Leap  Year  caused  ?  To  which  month  is 
the  odd  day  added  1  From  what  are  the  namua  of  the  days  derived  1 

6 


82  COMPOUND  [SECT.  VII, 

Tuesday  is  derived  from  Tuisco,  the  Saxon  god  of  war. 
Wednesday  is  derived  from  Woden,  a  deity  of  northern  Europe. 
TJiursday  is  from  Thor,  the  Danish  god  of  thunder,  storma,  <fec. 
Friday  is  from  Frig  a,  the  Saxon  goddess  of  beauty. 
Saturday  is  from  the  planet  Saturn,  to  which  it  was  dedicated 

1O3.  The  following  are  the  names  of  the  12  calendar 
months,  with  the  number  of  days  in  each : 


January, 

February, 

March, 

April, 

May, 

June, 

July, 

August, 

September, 

October, 

November, 

December, 


(Jan.)    the  first     month,  has  31  days. 


(Feb.)  "  second  "  "    28  " 

(Mar.)  "  third  "  "31  " 

(Apr.)  "fourth  "  "    30  " 

(May)  "  fifth  "  "31  " 

(June)  "  sixth  "  "    30  " 

(July)  "  seventh  "  "    31  " 

(Aug.)  "  eighth  "  "31  " 

(Sept.)  "  ninth  "  "    30  " 

(Oct.)  "  tenth  "  "31  " 

(Nov.)  "  eleventh  "  "    30  " 

(Dec.)  "  twelfth  "  "31  «' 

OBS.  1.  The  number  of  days  in  each  month  may  be  easily  re- 
membered from  the  following  lines  : 

"  Thirty  days  hath  September, 
April,  June,  and  November  ; 
February  twenty-eight  alone, 
All  the  rest  have  thirty-one ; 
Except  in  Leap  Year,  then  is  the  time, 
When  February  has  twenty-nine." 

2.  The  names  of  the  calendar  months  were  borrowed  from  the 
Romans,  and  most  of  them  had  a  fanciful  origin.  Thus, 

January  was  named  after  Janus,  a  Roman  deity,  who  WAS  sup- 
posed to  preside  over  the  year,  and  the  commencement  of  all 
undertakings. 

February  was  derived  fromfebrno,  a  Latin  word  which  signifies 
to  purify  by  sacrifice,  and  was  so  called  because  this  month  was 
devoted  to  the  purification  of  the  people. 


QUEST.— 103.  What  is  the  origin  of  the  narr.es  of  the  month*? 


ARTS.  103 — 105.]  NUMBERS.  88 

March  was  named  after  Mars,  the  Roman  god  of  war ;  and  was 
originally  the  first  month  of  the  Roman  year. 

April,  from  the  Latin  aperio,  to  open,  was  so  called  from  the 
opening  of  buds,  blossoms,  cfec.,  at  this  season. 

May  was  named  after  the  goddess  Maia,  the  mother  of  Mercury, 
to  whom  the  ancients  used  to  offer  sacrifices  on  ike  first  day  of 
this  month. 

June  was  named  after  the  goddess  Juno,  the  wife  of  Jupiter. 

July  was  so  called  in  honor  of  Julius  Ccesar,  who  was  born  in 
this  month. 

August  was  so  called  in  honor  of  Augustus  Ccesar,  a  Roman 
Emperor,  who  entered  upon  his  first  consulate  in  this  month. 

September,  from  the  Latin  numeral  septem,  seven,  was  so  called, 
because  it  was  originally  the  seventh  month  of  the  Roman  year. 
It  is  the  ninth  month  in  our  year. 

October,  from  the  Latin  octo,  eight,  was  so  called  because  it  was 
the  eighth  month  of  the  Roman  year. 

November,  from  the  Latin  novem,  nine,  was  so  called  because  it 
was  the  ninth  month  of  the  Roman  year. 

December,  from  the  Latin  decem,  ten,  was  so  called  because  it 
was  the  tenth  month  of  the  Roman  year. 

104.  The  year  is  also  divided  mtofour  seasons  of 
three  months  each,  viz:   Spring,   Summer,  Autumn  or 
Fall,  and  Winter. 

Spring  comprises  March,  April,  and  May ;  Summer, 
June,  July,  and  August ;  Autumn  or  Fall,  September, 
October,  and  November ;  Winter,  December,  Jan.  and  Feb. 

CIRCULAR  MEASyRE. 

105.  Circular  Measure  is  applied  to  the  divisions  of 
the  circle,  and  is  used  in  reckoning  latitude  and  longitude, 
and  the  motion  of  the  heavenly  bodies. 

60  seconds  (")  make  1  minute,  marked  ' 
60  minutes                   "      1  degree, 

30  degrees                   "      1  sign,  "       s. 

12  signs,  or  360°         "      1  circle,  "       c. 

QUKST.— 104.  Name  the  seasons.    105.  To  what  is  Circular  Measure  applied  1 


84 


COMPOUND 


[SECT.  VII. 


OBS.  J.  Circular  Measure  is 
c  ten  tailed  Angular  Measure, 
%nd  is  chiefly  used  by  astrono- 
mers, navigators,  and  surveyors. 

2.  The  circumference  of  every 
circle  is  divided,  or  supposed  to 
be  divided,  into  360  equal  parts, 
called  degrees,  as  in  the  sub- 
icined  figure. 

3.  Since  a  degree  is  yfcr  part 
of  the  circumference  of  a  circle, 
it  is    obvious    that    its    length 
must  depend  on  the  size  of  the  circle. 


270o 


MISCELLANEOUS  TABLE. 

1O6.  The  following  denominations  not  included   in 
the  preceding  Tables,  are  frequently  used. 


12  units 

12  dozen,  or  144 

12  gross,  or  1728 

20  units 

56  pounds 
100  pounds 

30  gallons 

200  Ibs.  of  shad  or  salmon 
196  pounds 
200  pounds 

14  pounds  of  iron,  or  lead 
21£  stone 
8  pigs 

OBS.  Formerly  112  pounds  were  allowed  for  a  quintal. 

QUEST. — Obs.  What  is  Circular  Measure  sometimes  called  ?  By  whom  ia  i| 
chiefly  used  ?  Into  what  is  the  circumference  of  every  circle  divided  ?  On 
what  does  the  length  of  a  degree  depend  ?  10G.  How  many  units  make  a 
dozen  ?  How  many  dozen  a  gross  1  A  great  gross  ?  How  many  units  maka 
a  score  ?  Pounds  a  flrjdn  ? 


make  1  dozen,  (doz.) 
"     I  gross. 
"     1  great  gross. 
"     1  score. 
"     1  firkin  of  butter. 
"     1  quintal  of  fish. 
"     1  bar.  of  fish  in  Mass. 

1  bar.  in  N.  Y.  and  Ct 

1  bar.  of  flour. 

1  bar.  of  pork. 

1  stone. 

1  P^ 
1  fother. 


ARTS.  106  —  108.]        NUMBERS.  85 

PAPER  AND  BOOKS. 

1OT.  The  terms,  folio,  quarto,  octavo,  &c.,  applied  to 
books,  denote  the  number  of  leaves  into  which  a  sheet 
0f  paper  is  folded. 

24  sheets  of  paper  make  1  quire. 

20  quires  "  1  ream. 

2  reams  "  1  bundle. 

5  bundles  "  1  bale. 

A  sheet  folded  in  two  leaves,  is  called  &  folio. 
A  sheet  folded  in  four  leaves,  is  called  a  quarto,  or  4te. 
A  sheet  folded  in  eight  leaves,  is  called  an  octavo,  or  8vo. 
A  sheet  folded  in  twelve  leaves,  is  called  a  duodecimo. 
A  sheet  folded  in  sixteen  leaves,  is  called  a  16  wo. 
A  sheet  folded  in  eighteen  leaves,  is  called  an  18  wo. 
A  sheet  folded  in  thirty-two  leaves,  is  called  a  39mo. 
A  sheet  folded  in  thirty-six  leaves,  is  called  a  3  6  wo. 
A  sheet  folded  in  forty-eight  leaves,  is  called  a  4  8  wo. 

1O8«  Previous  to  the  adoption  of  Federal  money  in 
1786,  accounts  hi  the  United  States  were  kept  in  pounds, 
shillings,  pence,  and  farthings. 

In  New  England  currency,  Virginia,  Ken-  i 

tucky,  Tennessee,  Indiana,  Illinois,  Mis-  >6  shil.  make  $1. 

Bouri,  and  Mississippi,  j 

In  New  York  currency,  North  Carolina,  ) 

8  shil  makc  &1- 


,  ) 
] 


Ohio,  and  Michigan, 
In   Pennsylvania  currency,   New  Jersey,  )  _, 

Delaware,  and  Maryland,  \  7s'  6d'  make  $1 

In  Georgia  currency,  and  South  Carolina,      4s.  8d.  make  $1. 
In  Canada  currency,  and  Nova  Scotia,  5  shil.  make  $1. 

QUEST.—  107.  When  a  sheet  of  paper  is  folded  in  two  leaves,  what  is  it 
called  ?  When  in  four  leaves,  what  ?  When  in  eight  ?  In  twelve  ?  In 
sixteen  1  In  eighteen  1  In  thirty-six  ?  108.  Previous  to  the  adoption  of  Fed- 
eral Money,  in  what  were  accounts  kept  in  the  U.  S.  ?  How  many  shillings 
make  a  dollar  in  N.  E.  c\irrency  1  In  N.  V.  currency  ^  !\n  Penn.  currency  1 
In  Georgia  currency  1  In  Canada  currency  7 


86  COMPOUND  [SECT.  VIL 

OBS.  At  the  time  Federal  money  was  adopted,  the  colonial  cur* 
rency  or  bills  of  credit  issued  by  the  colonies,  had  more  or  less  de- 
preciated in  value :  that  is,  a  colonial  pound  was  worth  less  than  a 
pound  Sterling;  a  colonial  shilling,  than  a  shilling  Sterling,  &e. 
This  depreciation  being  greater  in  some  colonies  than  in  others, 
gave  rise  to  the  different  values  of  the  State  currencies. 

ALIQUOT    PARTS    OF    $1    IN    FEDERAL   MONEY. 


50     cents  = 

33i  cents  = 

25     cents  = 

20     cents  = 

16|  cents  = 


12  J  cents  ~ 

10     cents  =* 

8|  cents  = 

6|  cents  = 

5     cents  = 


PARTS  OF  $1  IN  NEW  YORK  CURRENCY. 


4  shillings  == 

2  shil.  8  pence      = 
2  shillings  = 


1  shil.  4  pence   = 
1  shilling  = 

6  pence  = 


OBS.  1.  In  New  York  currency,  it  will  be  seen,  (Art.  108,^  that 
A  six-pence,  written    6d.  =     6^  cents, 

A  shilling,  "          Is.  =  12£      " 

One  (shil.)  and  6  pence,         «       1/6.  =  18J      " 
Two  shillings,  "          2s.  =  25        " 

PARTS  OF  $1  IN  NEW  ENGLAND  CURRENCY. 

3  shillings  =  $-£•          1  shilling  =  $f 

2  shillings  =  $ -J-          9  pence  =  $fc 

I  shil.  and  6d.      =  $i          6  pence  =  $fV 

OBS.  2.  In  New  England  currency,  it  will  be  seen,  that 
A  four-pence-half-penny,  written  4£d.  =    6£  cents. 
A  six-pence,  "          Gd.  =     8£      " 

A  nine-pence,  "          9d.  =  12^      {C 

A  shilling,  "  Is.  =  16§      f« 

One  (shil.)  and  six-pence,      "         1/6.  =  25        " 
Two  shillings,  "          2s.  =  33  J      " 

QITEST. — What  are  the  aliquot  parts  of  $1  in  Federal  Money  7    In  New  York 
currency  7    In  Now  England  currency  7    What  are  the  aliquot  parts  of  a  pou 
Sterling  7    Of  a  shilling  7 


ART.  108.] 


NUMBERS. 


87 


ALIQUOT    PARTS    OF    STERLING    MONEY. 

Parts  of  £1. 

Parte  o/  Is. 

10  shil.     =    £i 

6     pence  =  -J-    shil. 

6s.  8d.    =    £i 

4     pence  =  -£•    shil. 

5  shil.     =    £i 

3     pence  =  •}•    shiL 

4  shil.     =    £i 

2     pence  =  -J-    shil. 

3s.  4d.    ~    £| 

1£  pence  =  i    shil. 

2s.  6d.    =    £i 

1     penny  =  iV  shil. 

2  shil.     =    £11(r 

1     far.  =  i  penny. 

Is.  8d.    —    £-iV 

2     far.  =  -J-  penny. 

1  shil.     =    £-21o 

3     far.  =  f  penny. 

ALIQUOT   PARTS    OR  A    TON. 


10  hund.  lbs.=i  ton. 
5  hund.  lbs.=-J-  ton. 
4  hund.  lbs.=-£-  ton. 


2  hund.  2  qrs.=i  ton. 
2  hund.  Ibs.  =-fo  ton. 
1  hund.  Ibs.  —^  ton. 


ALIQUOT    PARTS    OF    A    POUND    AVOIRDUPOIS. 


8  ounces =%  pound. 
4  ounces =-J-  pound. 


2  ounces =i  pound. 
1  ounce  =iV  pound. 


ALIQUOT    PARTS    OF    TIME. 

Parts  of  1  year. 

Parts  of  1  month. 

6    months  =•£  year. 

15  days=i-     month. 

4    months  =%  year. 

10  day  s==^-     month. 

3    months  =-J-  year. 

6  days=£     month. 

2    months  =-J-  year. 

5  days=-J-     month. 

1J-  month  =-J-  year. 

3  days=-iV  month. 

1-J-  month  =|  year. 

2  days^iV  month. 

1    month  ==-i12  year. 

1  day  =-sV  month. 

QUEST.— How  many  shillings  in  half  a  pound  Ster.  ?  In  a  fourth ?  A  fifth  ? 
A  tenth  ?  A  twentieth  V  How  many  pence  in  half  a  shilling  ?  In  a  third  ?  A 
fourth?  A  sixth 7  A  twelfth?  How  many  hundreds  in  half  a  ton?  In  a 
fourth  ?  A  fiftn  ?  A  tenth  ?  How  many  ounces  in  half  a  pound  ?  In  a  fourth  ? 
An  eighth?  A  sixteenth?  How  many  months  in  half  a  year?  la  a  third? 
A  fourth?  A  sixth?  A  twelfth  7 


88  FEDERAL  MONEY.  [SECT.  VIIL 

SECTION    VIIL 

FEDERAL   MONEY. 

1 1 0.  Accounts  in  the  United  States  are  kept  in  dol- 
lars, cents,  and  mills.  Eagles  are  expressed  in  dollars,  and 
dimes  in  cents.  Thus,  instead  of  4  eagles,  we  say,  40  dol- 
lars ;  instead  of  5  dimes,  we  say,  50  cents,  &c. 

Ill*  Dollars  are  separated  from  cents  by  placing  a 
point  or  separatrix  ( . )  between  them.  Hence, 

112.  To  read  Federal  Money. 

Call  all  the  figures  on  the  left  of  the  point,  dollars  ;  the 
first  two  figures  on  the  right  of  the  point,  are  cents  ;  the 
third  figure  denotes  mills  ;  the  other  places  on  the  right  are 
parts  of  a  mill.  Thus,  $5.2523  is  read,  5  dollars,  25 
cents,  2  mills,  and  3  tenths  of  a  mill. 

OBS.  1.  Since  two  places  are  assigned  to  cents,  when  no  cents 
are  mentioned  in  the  given  number,  two  ciphers  must  be  placed 
before  the  mills.  Thus,  5  dollars  and  7  mills  are  written  $  5.007. 

2.  If  the  given  number  of  cents  is  less  than  ten,  a  cipher  must 
always  be  written  before  them.  Thus,  8  cents  are  written  .08,  <fcc. 

1.  Read  the  following  expressions:  $83.635  ;  $75.50. 
$126.607;  $268.05;  $382.005;  $2160. 

2.  Write  the  following  sums :  Sixty  dollars  and  fifty 
cents.     Seventy-five  dollars,  eight  cents,  and  three  mills. 
Forty-eight  dollars   and   seven    mills.     Nine   cents.     Six 
cents  and  four  mills. 

QUEST.— 88.  What  is  Federal  Money?  What  are  its  denominations?  Re- 
cite the  Table.  110.  How  are  accounts  kept  iu  the  United  States?  How  are 
Eagles  expressed ?  Dimes?  111.  How  are  dollars  distinguished  from  cents 
and  mills  ?  112.  How  do  you  read  Federal  Money  ?  Obs.  How  many  places 
are  aseigned  to  cents  ?  When  the  number  of  cents  is  less  than  ten,  what  must 
be  done  ?  When  no  cents  are  mentioned,  what  do  you  do  ? 


ARTS.  110  —  113.]      FEDERAL  MONEY.  89 

REDUCTION   OF  FEDERAL   MONEY. 

CASE    I. 
Ex.  1,  How  many  cents  are  there  in  65  dollars? 

Suggestion.  —  Since  in  1  dollar  there  are      Operation. 
100  cents,  in  65  dollars  there  are  65  times  as     65 
many.     Now,  to  multiply  by  10,  100,  &c.,we 


annex  as  many  ciphers  to  the  multiplicand,     6500  cents. 
as  there  are  ciphers  in  the  multiplier.     (Art.  45.)     Hence, 

113.  To  reduce  dollars  to  cents,  annex  TWO  ciphers. 
To  reduce  dollars  tc  mills,  annex  THREE  ciphers. 
To  reduce  cents  to  mills,  annex  ONE  cipher. 

OBS.  To  reduce  dollars  and  cents  to  cents.,  erase  the  sign  of  dollars 
and  the  separatrix.  Tfefes,  $25.36  reduced  te  cents,  become  2536 
cents. 

2.  Reduce  $4  to  cents.  Ans.  400  cents. 

3.*  Reduce  $15  to  cents.  7.  Reduce  $96  to  mills. 

4.  Reduce  $27  to  cents.  8.  Reduce  $12.23  to  cents. 

5.  Reduce  $85  to  cents.  9'.  Reduce  $86.86  to  cents. 

6.  Reduce  $93  to  cents.  10.  Reduce  $9.437  to  mills. 

CASE    II. 
1.  In  2345  cents,  how  many  dollars  ? 

Suggestion.  —  Since  100  cents  make  1  dol-  Operation. 
lar,  2345  cents,  will  make  as  many  dollars  1|00)23|45 
as  100  is  contained  times  in  2345.  Now  to  Ans.  $23.45 
divide  by  10,  100,  &c.,  we  cut  off  as  many 
figures  from  the  right  of  the  dividend  as  there  are  ciphera 
in  the  divisor.  (Art.  67.)  Hence, 


QUEST.— 113.  Ho\v  are  dollars  reduced  to  cents?    Dollars  to  mills  ?    Centa 
to  mills  ?    Obs.  Dollars  and  cents  to  cents  ? 


90  FEDERAL  MONET.  [SECT.  VIIL 

114*  To  reduce  cents  to  dollars. 

Point  off  TWO  figures  on  the  right  ;  the  figures  remain* 
ing  on  the  left  express  dollars  ;  the  two  pointed  off,  cents. 

1  1  5.  »  To  reduce  mills  to  dollars. 

Point  off  THREE  figures  on  the  right  ;  the  remaining 
figures  express  dollars  ;  the  first  two  on  the  right  of  the 
point,  cents  ;  the  third  one,  mills. 

116*  To  reduce  mills  to  cents. 

Point  off  ONE  figure  on  the  right,  and  the  remaining 
figures  express  cents  ;  the  one  pointed  off,  mills. 

2.  Reduce  236  cts.  to  dolls.  3.  Reduce  21  63  cts.  to  dolls. 
4.  Reduce  865  mills  to  dolls.  5.  Reduce  906  mills  to  cts. 
6.  Reduce  2652  cts.  to  dolls.  7.  Redfce  3068  cts.  to  dolls. 

ADDITION  OF  FEDERAL  MONEY. 
Ex.  1.  What  is  the  sum  of  $8.125,  $12.67,  $3.098,  $11  ? 


Suggestion.—  Write  the  dollars   under 

dollars,   cents   under   cents,    mills   tinder  1267 

mills,  and  proceed  as  in  Simple  Addition.  3  098 

From  the  right  of  the  amount  point  off  11.00 

three  figures  for  cents  and  mills.  Ans.  $34.893 

117*  Hence,  we  derive  the  following  general 

RULE  FOR  ADDING  FEDERAL  MONEY. 

Write  dollars  under  dollars,  cents  under  cents,  mills 
under  mills,  and  add  each  column,  as  in  simple  numbers. 

From  the  right  of  the  amount,  point  off  as  many  figures 
for  cents  and  mills,  as  there  are  places  of  cents  and  mills 
in  either  of  the  given  numbers. 

QUEST.-—  114.  How  are  cents  reduced  to  dollars?   115.  Mills  to  dollars?  117 
How  do  you  add  Federal  Money?    How  point  off  the  amount? 


ARTS.  114 — 11*7.]     FEDERAL  MONEY.  91 

OBS.  If  either  of  the  given  numbers  have  no  cents  expressed, 
•upply  their  place  by  ciphers. 


(2.) 

$375.037 

(3.) 

$4869.45 

$760.275 

(5.) 

$4607.375 

60.20 

344.00 

897.008 

.   896.084 

843.462 

6048.07 

965.054 

95.873 

(6.) 

$782.206 

$609.352 

(8.) 
$2903.76 

(9.) 

$4668.253 

84.60 

830.206 

453.06 

430.064 

379.007 

408.07 

25.89 

307.60 

498.015 

631.107 

6842.07 

7452.349 

10.  What  is  the  sum  of  $63.072,  $843.625,  and  $71:60 ! 

11.  Add  $873.035,  $386.23,  $608.938,  $169.176. 

12.  Add  463  dolls.  7  cts. ;  248  dolls.  15  cts. ;  169  dolls. 
9  cts.  7  mills. 

13.  Add  89  dolls.  8  cts.;  97  dolls.  10  cts.  3  mills;  40 
dolls.  6  cts. ;  75  dolls. 

14.  Add  365  dolls.  20  cts.  2  mills;   68  dolls.  6  cts.  5 
mills  ;  7  cts.  3  mills ;  286  dolls. ;  80  dolls.  6  mills ;  30  dolls. 
15  cts. 

15.  A  man  bought  a  cow  for  $16.375,  a  calf  for  $4.875, 
and  a  ton  of  hay  for  $13.50  :  how  much  did  he  pay  foi 
the  whole  ? 

16.  A  lady  paid  $23  for  a  cloak,  $7.625  for  a  hat,  $25.75 
for  a  muff,  and  $18  for  a  tippet :  how  much  did  she  pay 
for  all  ? 

17.  A  farmer  sold  a  cow  for  $16.80,  a  calf  for  $4.08,  a 
horse  for  $78,  and  a  yoke  of  oxen  for  $63.18  :  how  much 
did  he  receive  for  all  ? 


QUEST.— Obs.  When  any  of  the  given  numbers  have  no  cents  expressed, 
how  ia  their  place  supplied? 


92  FEDERAL    MONEY.  [SECT.  VIIL 

SUBTRACTION  OF  FEDERAL  MONEY. 

Ex.  1.  What  is  the  difference  between  $845.634,  and 
$86.087  ? 

Suggestion. — Write  the  less  number         Operation. 
under  the  greater,  dollars  under  dollars,  $845.634 

&c.,  then  subtract,  and  point  off  the  an-  86.087 

swer  as  in  addition  of  Federal  Money.         Ans.  $759.547. 

118*  Hence,  we  derive  the  following  general 

EULE   FOR  SUBTRACTING   FEDERAL   MONEY. 
Write  the  less  number  under  the  greater,  with  dollars 
under  dollars,  cents  under  cents,  and  mills  under  mills  ; 
then  subtract,  and  point  off  the  answer  as  in  addition 
of  Federal  Money. 


(2.) 
From  $856.453 
Take  $387.602 

(3.) 

$960.78 
$463.05 

(4.) 

$605.607 
78.36 

(5.) 
$6243.760 
327.053 

(6.) 
From  965.005 
Take   87.85 

840.000 
378.457 

(8.) 
483.853 

48.75 

(9.) 
4265.76 
2803.98 

10.  From  $86256.63  take  $4275.875  ? 

11.  From  $100250,  take  $32578.867  1 

12.  From  1  dollar,  subtract  11  cents. 

13.  From  3  dolls.  6  cts.  7  mills,  take  75  cents. 

14.  From  110  dolls.  8  mills,  take  60  dolls,  and  8  cents. 

15.  From  607  dolls.  7  cents,  take  250  dolls,  and  3  cts. 

16.  A  lad  bought  a  cap  for  $2.875,  and  paid  a  five- 
dollar-bill  :  how  much  change  ought  he  to  receive  back  ? 

17.  Henry  has  $7.68,  and  William  has  $9.625  :  how 
much  more  has  the  latter  than  the  former  ? 

18.  From  $865275.60,  take  $340076.875. 

QUEST.— 106.  How  do  you  subtract  FederaJ  Money  1    How  point  off  tht» 
nnswer  1 


ARTS.  118,  119.]        FEDERAL  MONEY.  93 

MULTIPLICATION  OF  FEDERAL  MONEY. 

Ex.  1.  What  will  3  caps  cost,  at  $1.625  apiece  ? 

Suggestion. — Since  1  cap  costs  $1.625,  Operation. 
8  caps  will  cost  3  times  as  much.  We  *^  ^25 

therefore  multiply  the  price  of  1  cap  by  3,  3 

the   number  of  caps,  and  point  off  three     Ans.  $4.875. 
places  for  cents  and  mills.     Hence, 

111).  When  the  multiplier  is  a  whole  number,  we  have 
the  following 

RULE   FOR   MULTIPLYING   FEDERAL   MONEY. 

Multiply  as  in  simple  numbers,  and  from  the  right  of 
the  product,  point  off  as  many  figures  for  cents  and  mills, 
as  there  are  places  of  cents  and  mills  in  the  multiplicand. 

OBS.  1.  In  Multiplication  of  Federal  Money,  as  well  as  in  simple 
numbers,  the  multiplier  must  always  be  considered  an  abstract 
number. 

2.  In  business  operations,  when  the  mills  are  5  or  over,  it  is 
customary  to  call  them  a  cent ;  when  under  5,  they  are  disregarded. 

(2.)  (3.)  (4.)  (5.) 

Multiply    $633.75       $805.625       $879.075       $9071.26 
By  8  9  24  37 


(6.) 

(7.) 

(8.) 

(9.) 

Multiply  $4063.36 

$5327.007 

$6286.69 

$8265.68 

By                       63 

86 

123 

264 

10.  What  cost  8  melons,  at  17  cents  apiece  ? 

11.  What  cost  12  lambs,  at  87  cents  apiece  ? 

12.  What  cost  8  hats,  at  $3.875  apiece  1 

13.  At  $8.75  a  yard,  what  will  9  yards  of  silk  come  to! 

14.  At  $1.125  apiece,  what  will  11  turkeys  cost? 

QUEST.— 119.  How  do  you  multiply  Federal  Money  ?  How  point  off  the 
product?  065.  What  must  the  multiplier  always  be  considered  ?  When 
the  mills  are  5,  or  over,  what  is  it  customary  to  call  them  ?  When  lees  than 
5  what  may  be  done  with  them  ? 


94  FEDERAL   MONEY.  [SECT.  VIIL 

15.  At  $2.63  apiece,  what  will  15  chairs  come  tof 

16.  What  costs  25  Arithmetics,  at  37-^  cents  apiece! 

17.  "What  cost  38  Readers,  at  62|-  cents  apiece? 

18.  What  cost  46  over-coats,  at  $25.68  apiece  ? 

19.  What  cost  69  oxen,  at  $48.50  a  head  ? 

20.  At  $23  per  acre,  what  cost  65  acres  of  land  ? 

21.  At  $75.68  apiece,  what  will  56  horses  come  to  ? 

22.  At  7-J  cents  a  mile,  what  will  it  cost  to  ride  100 
miles  ? 

23.  A  farmer  sold  84  bushels  of  apples,  at  87-J-  cents  per 
bushel :  what  did  they  come  to  1 

24.  If  I  pay  $5.3 7|-  per  week  for  board,  how  much  will 
it  cost  to  board  52  weeks  ? 

DIVISION  OF  FEDERAL  MONEY. 
Ex.  1.  If  you  paid  $18.876  for  3  barrels  of  flour,  how 
much  was  that  a  barrel  ? 

Suggestion. — Since  3  barrels  cost  $18.- 
876,  1  barrel  will  cost  1  third  as  much, 
We  therefore  divide  as  in  simple  division, 
and  point  on  three  places  for  cents  and 
mills,  because  there  are  three  in  the  dividend.  Hence, 

1 2O.  When  the  divisor  is  a  whole  number,  we  have 
the  following 

RULE   FOR  DIVIDING   FEDERAL   MONEY. 
Divide  as  in  simple  numbers,  and  from  the  right  of  the 
quotient,  point  off  as  many  figures  for  cents  and  mills,  as 
there  are  places  of  cents  and  mills  in  the  dividend. 

OBS.  "When  the  dividend  contains  no  cents  and  mills,  if  there 
is  a  remainder  annex  three  ciphers  to  it ;  then  divide  a&  before, 
and  point  off  three  figures  in  the  quotient. 

QUKST.— 320.  How  do  you  divide  Federal  Money  ?  How  point  off  the 
quotient  ?  Obs>  When  the  dividend  contains  no  cents  and  mills,  how  f  roceed  ? 


AAT.  120.]  FEDERAL   MONEY  95 

Note. — For  a  more  complete  development  of  multiplication  and 
division  of  Federal  Money,  the  learner  is  referred  to  the  author's 
Practical  and  Higher  Arithmetics. 

When  the  multiplier  or  divisor  contain  decimals,  or  cents  and 
mills,  to  understand  the  operation  fully,  requires  a  thorough 
knowledge  of  Decimal  Fractions,  a  subject  which  the  limits  of  this 
work  will  not  allow  us  to  introduce, 

(2.)  (3.)  (4.) 

6)  $856.272.  8)  $9567.648.  9)  $7254,108. 

5.  Divide  $868.36  by  27.       6.  Divide  $3674.65  by  38. 

7.  Divide  $486745  by  49.      8.  Divide  $634.075  by  ofi. 

9.  Divide  $6634.25  by  60.   10.  Divide  $5340.73  by  78 
11.  Divide  $7643.85  by  83.   12.  Divide  $4389.75  by  89. 
13.  Divide  $836847  by  94.  14.  Divide  $94321.62  by  97. 

15.  A  man  paid  $2563.84  for  63  sofas  :  what  was  that 
apiece  ? 

16.  A  miller  sold  86  barrels  of  flour  for  $526.50 :  how 
much  was  that  per  barrel  ? 

17.  If  a  man  pays  $475.56  for  65  barrels  of  pork,  what 
is  that  per  barrel  ? 

18.  A  man  paid  $1875.68  for  93  stoves:  how  much 
was  that  apiece  ? 

19.  If  $2682.56  are  equally  divided   among  100  men, 
how  much  will  each  receive  ? 

20.  A  cabinet-maker  sold  116  tables  for  $968.75  :  how 
much  did  he  get  apiece  ? 

21.  A  farmer  sold  168  sheep  for  $465  :  how  much  did 
he  receive  apiece  for  them  ? 

22.  A  miller  bought  216  bushels  of  wheat  for  $375.50  : 
how  much  did  he  pay  per  bushel  ? 

23.  If  $2368.875  were  equally  divided  among  348  per- 
sons, how  much  would  each  person  receive  ? 


98  REDUCTION.  [SECT.  IX. 

SECTION    IX 

REDUCTION. 

ART.  121*  REDUCTION  is  the  process  of  changing 
Compound  Numbers  from  one  denomination  into  another 
without  altering  their  value. 

REDUCING  HIGHER  DENOMINATIONS  TO  LOWER. 

122.  Ex.  1.  Reduce  £2,  to  farthings. 

Suggestion.  —  First    reduce    the  Operation. 

given   pounds  (2)    to   shillings,  by  £2 

multiplying   them   by  20,  because  20s.  in  £l. 

20s.  make  £l.     Next   reduce  the  40  shillings, 

shillings  (40)  to  pence,  by  multi-  I2d.  in  Is. 

plying  them  by   12,  because   12d.  480  pence. 

make  Is.     Reduce  the  pence  (480)  _1 far- in  ld- 

to  farthings,  by  multiplying  them    ^ns-  192°  farthings. 
by  4,  because  4  far.  make  Id. 

2.  Reduce  £l,  2s.  4d.  and  3  far.  to  farthings. 

Suggestion.  —  In    this     example  Operation. 

there  are  shillings,  pence,  and  far-  £    *•  d.  far. 

things.     Hence,  when  the   pounds  *    2.  4     3 

are   reduced  to  shillings,  the  given 
shillings  (2)  must  be  added  men-  itd^iJT 

tally  to   the  product.     When  the 

'  x  ,       ,  ,  268  pence, 

shillings  are  reduced  to  pence,  the  4  f  -  '     id 

given  pence  (4)  must  be  added;    ^  ^^ ^      ' 

and  when  the  pence  are  reduced  to 

farthings,  the  given  farthings  (3)  must  be  added. 

Q,UEST. — 121.  What  is  reduction?  122.  Ex.  1.  How  reduce  pounds  to  shil- 
lings? Why  multiply  by  20  ?  How  are  shillings  reduced  to  pence?  Why  ? 
How  pence  to  farthings  1  Why  ? 


ARTS.  121 — 124.]          REDUCTION.  0Y 

OBS.  lu  these  examples  it  is  required  to  reduce  higher  denomi 
nations  to  lower,  as  pounds  to  shillings,  shillings  to  pence,  <fcc. 

123*  The  process  of  reducing  higher  denominationa 
to  lower,  is  usually  called  Reduction  Descending. 

It  consists  in  successive  multiplications,  and  may  with 
propriety  be  called  Reduction  by  Multiplication. 

124*  From  the  preceding  illustrations  we  derive  the 
following 

RULE  FOB  REDUCTION  DESCENDING. 

Multiply  the  highest  denomination  given  by  the  num- 
ber required  of  the  next  lower  denomination  to  make  ONE 
of  this  higher,  and  to  the  product,  add  the  given  number  of 
this  lower  denomination. 

Proceed  in  this  manner  with  each  successive  denomina- 
tion, till  you  come  to  the  one  required. 

EXAMPLES. 

3.  Reduce  4  miles,  2  fur.,  8  rods  and  4  feet  to  feet. 

Operation. 

Suggestion. — Having  reduced  the  m.fur.  r.  ft. 

miles  and  furlongs  to  rods,  we  have 
1368  rods.     We  then  multiply  by  —  . 

10-J-,  because  16^-  feet  make  1  rod.  ,Q 

(Art.  94.)     Now   16 J  is  a  mixed          2)1368  rods, 
number;  we  therefore  first   multi-  jgi 

ply   by   the   whole    number   (16),  8212 

then  by  the  fraction  (-£•),  and  add  1368 

the  products  together.     (Art.  84.)  684 

Ans.  22576  feet. 


QUEST.— 123.  What  is  reducing  compound  numbers  to  lower  denominations 
usually  called?  Which  of  the  fundamental  rules  is  employed  in  reduction 
descending?  124.  What  is  the  rule  for  Reduction  Defending  ? 


98  REDUCTION.  |  SECT.   IX 

4.  In  £5, 16s.  7d.,  how  many  farthings  ?  Ans.  5596  far 

5.  In  £18  how  many  pence? 

6.  In  £23,  9s.,  how  many  shillings  ? 

7.  In  17s.  2d.  3  far.,  how  many  farthings? 

8.  Reduce  5  Ibs.  6  oz.  Troy  weight,  to  grains. 

Ans.  31680  grs. 

9.  Reduce  13  Ibs.  Troy,  to  ounces. 

'     10.  Reduce  4  Ibs.  3  oz.  Troy,  to  penny  weights. 

11.  Reduce  15  oz.  6  pwts.  4  grs.  Troy,  to  grain*. 

12.  In  2  cwt.  3  qrs.  7  Ibs.  4  .oz.  3  drams,  avoirdupow 
weight,  how  many  drams?     Ans.  72259  drams. 

13.  In  13  Ibs.  4  oz.  avoirdupois,  how  many  ounces? 

14.  In  2  qrs.  17  Ibs.  avoirdupois,  how  many  pounds  f 

15.  In  6  Ibs.  12  oz.  avoirdupois,  how  many  drams? 

16.  In  12  cwt.  1  qr.  6  Ibs.  avoir.,  how  many  ounces  * 

17.  In  16  miles,  how  many  rods? 

18.  In  28  rods  and  2  feet,  how  many  inches  ? 

19.  In  19  fur.  4  rods  and  2  yds.,  how  many  feet  ? 

20.  In  25  leagues  and  2  m.,  how  many  rods  ? 

21.  Reduce  14  yards  cloth  measure  to  quarters. 

22.  Reduce  21  yards  2  quarters  to  nails. 

23.  Reduce  17  yards  3  quarters  2  nails,  to  nails. 

24.  How  many  quarts  in  23  gallons,  wine  measure  ? 

25.  How  many  gills  in  30  gallons  and  2  quarts? 

26.  How  many  gills  in  63  gallons  ? 

27.  How  many  quarts  in  41  hogshead*  ? 

28.  How  many  pecks  in  45  bushels  ? 

29.  How  many  pints  in  3  pecks  and  2  quarts  ? 

30.  How  many  quarts  in  52  bu.  and  2  peck*  ? 

31.  How  many  hours  in  15  weeks? 

32.  How  many  minutes  in  25  days  ? 
83.  How  many  seconds  in  265  hours  ? 

34.  How  many  minutes  in  52  weeks  ? 

35.  How  many  seconds  in  68  days  ? 


ARTS.  125,  126.]  REDUCTION.  99 

REDUCING  LOWER  DENOMINATIONS  TO  HIGHER. 

125.  Ex.  1.  Reduce  1920  farthings  to  pounds. 

Suggestion. — First  reduce  the  given  far-     Operation. 
things  (1920)   to  pence,  the  next  higher        4)1920  far, 
denomination,  by  dividing  them  by  4,  be-        12)480d. 
cause  4  far.  make  Id.     Next  reduce  the          20)40s. 
pence  (480)  to  shillings,  by  dividing  them  £2  Ans. 

by  12,  because  12d.  make  Is.     Finally,  re- 
duce the  shillings  (40)  to  pounds,  by  dividing  them  by  20, 
because  20s.  make  £l.     The  answer  is  £2.     That  is,  1920 
far.  are  equal  to  £2. 

2.  In  1075  farthings,  how  many  pounds? 

Suggestion. — In    dividing   the  Operation. 

given  farthings  by  4,  there  is  a         4)1075  far. 
remainder  of  3  far.,  which  should         12)268d.  3  far.  over, 
be  placed  on  the  right.     In  di-  20)22s.  4d.  over, 

viding   the  pence  (268)   by   12,  £1,  2s.  over, 

there  is  a  remainder  of  4d.,  which  Ans.  £l,  2s.  4d.  3  far. 
should  also  be  placed  on  the 

right.  In  dividing  the  shillings  (22)  by  20,  the  quotient 
is  £l  and  2s.  over.  The  last  quotient  with  the  several 
remainders  is  the  answer.  That  is,  1075  far.  are  equal  to 
£1,  2s.  4d.  3  far. 

OBS.  In  the  last  two  examples,  it  is  required  to  reduce  lower  de- 
nominations to  higher,  as  farthings  to  pence,  pence  to  shillings,  &c. 
The  operation  is  exactly  the  reverse  of  that  in  Reduction  Descending. 

126*  The  process  of  reducing  lower  denominations  to 
higher  is  usually  called  Reduction  Ascending. 

It  consists  in  successive  divisions,  and  may  with  propri- 
ety be  called  Reduction  by  Division. 

QUFST.— 125.  Ex.  1.  How  are  farihings  reduced  to  pence  ?  Why  divide  by  4 1 
How  reduce  pence  to  shillings  1  WLy?  How  shillings  to  pounds?  Why? 
120.  What  is  reducing  compound  numbers  to  higher  denominations  usually 
called '«  Which  ol  the  fundamental  rules  ia  employed  in  Reduction  Ascending  ? 


100  REDUCTION.  [SECT.  IX. 

127.  From  the  preceding  illustrations  we  derive  the 
following 

RULE   FOR  REDUCTION   ASCENDING. 

Divide  the  given  denomination  ~by  that  number  which  it 
takes  of  this  denomination  to  ma  fee  ONE  of  the  next  higher. 
Proceed  in  this  manner  with  each  successive  denomination) 
till  you  come  to  the  one  required.  The  last  quotient,  with 
tlie  several  remainders,  will  be  the  answer  sought. 

128*  PROOF. — Reverse  the  operation;  that  is,  reduce 
back  tlie  answer  to  the  original  denominations,  and  if  the 
result  correspond  with  the  numbers  given,  the  work  is  right. 

OBS.  Each  remainder  is  of  the  satne  denomination  as  the  divi- 
dend from  which  it  arose.  (Art.  51,  Obs.  2.) 

EXAMPLES. 

3.  In  429  feet,  how  many  rods  ?  Operation, 
Suggestion. — We  first  reduce  the  feet         3  )429  feet. 

to  yards,  then  reduce  the  yards  to  rods  5i-)143  yds. 

by   dividing   them   by  5-J-.      (Art.  86.)  2         2 

Or,  we  may  divide  the  given  feet  by  11  )286 

16£,  the  number  of  feet  in  a  rod,  and  the  Ans.  26  rods, 
quotient  will  be  the  answer. 

Proof. 

We  first  reduce  the  rods  back  to  yards,  26  rods. 

(Art.  84,)  then  reduce  the  yards  to  feet.  5^ 

The  result  is  429  feet,  which  is  the  same  130 

as  the  given  number  of  feet.  13 

Or,  we  may  multiply  the  26  by  16i,  143  yds. 

and  the  product  will  be  429.  3 

429  feet. 

4.  Reduce  256  pence  to  pounds.     Ans.  £l,  Is.  4d. 

5.  Reduce  324  pence  to  shillings. 

QUEST. — 127,  What  is  the  rule  for  reduction  ascending  ?  328,  Hovr  is  re- 
duction proved  1  Obs.  Of  what  denomination  is  each  remainder  7 


ARTS.  127 — 129.]     REDUCTION.  101 

6.  Reduce  960  farthings  to  shillings. 

7.  Reduce  1250  farthings  to  pounds. 

8.  In  265  ounces  Troy  weight,  how  many  pounds  ? 

9.  In  728  pwts.,  how  many  pounds  Troy?       % 

10.  In  54'8  grains,  how  many  ounces  Troy? 

11.  In  638  oz.  avoirdupois  weight,  how  many  pounds? 

12.  In  736  Ibs.  avoirdupois,  how  many  quarters? 

13.  In  1675  oz.  avoirdupois,  how  many  hundred  weight  ? 

14.  In  1000  drams  avoirdupois,  how  many  pounds? 

15.  In  4000  Ibs.  avoirdupois,  how  many  *,ons? 

16.  How  many  yards  in  865  inches  ? 

17.  How  many  rods  in  1000  feet? 

18.  How  many  miles  in  2560  rods  ? 

19.  How  many  miles  in  3261  yards  ? 

20.  How  many  leagues  in  2365  rods  ? 

EXAMPLES    IN   REDUCTION    ASCENDING    AND    DESCENDING. 

129*  In  solving  the  following  examples,  the  pupil 
tnust  first  consider  whether  the  question  requires  higher 
denominations  to  be  reduced  to  lower,  or  lower  denomina- 
tions to  higher.  Having  settled  this  point,  he  "ill  find  no 
difficulty  in  applying  the  proper  rule. 

FEDERAL   MONEY.     (ART.  88.) 

1.  In  3  dollars  and  16  cents,  how  many  cents  ? 

2.  In  81  cents  and  2  mills,  how  many  mills? 

3.  In  245  cents,  how  many  dollars? 

4.  In  321  mills,  how  many  dimes? 

5.  In  95  eagles,  how  many  cents  ? 

6.  In  160  dollars,  how  many  cents  ? 

7.  In  317  dollars,  how  many  dimes? 

8.  In  4561  mills,  how  many  dollars? 

9.  In  8250  cents,  how  many  eagles  ? 

10.  In  61  dolls.,  12  cts.,  *md  3  mills,  how  many  milk? 


02  REDLCTION.  [SECT.    IX. 

STERL  .NG  MONEY.    (ART.  90.) 
11.  Keduce  £17,  16s.  to  shillings. 
12:  Reduce  19s.  6d.  2  far.  to  farthings. 

13.  ^Reduce  1200  pence  to  pounds. 

14.  Reduce  3626  farthings  to  shillings. 

15.  Reduce  £19  to  farthings. 

16.  Reduce  2880  farthings  to  shillings. 

17.  Reduce  £21,  3s.  6d.  to  pence. 

18.  Reduce  3721  farthings  to  pounds. 

TROY   WEIGHT.     (ART.  91.) 
.19.  In  7  Ibs.,  how  many  ounces  ? 

20.  In  9  Ibs.  2  oz.,  how  many  pennyweights  ? 

21.  In  165  oz.,  how  many  pounds? 

22.  In  840  grains,  how  many  ounces  ? 

23.  In  3  Ibs.  5  oz.  2  pwts.  7  grs.,  how  many  grains? 

24.  In  6860  grains,  how  many  pounds? 

AVOIRDUPOIS   WEIGHT.     (ART.  92.) 

25.  In  200  oz.,  how  many  pounds  ? 

26.  In  261  Ibs.,  how  many  ounces? 

27.  In  3  tons,  2  cwt.,  how  many  pounds? 

28.  In  1  cwt.  2  qrs.,  how  many  ounces  ? 

29.  In  1000  oz.,  how  many  pounds? 

30.  In  4256  Ibs.,  how  many  tons  ? 

APOTHECARIES'   WEIGHT.     (ART.  W.) 

31.  Reduce  45  pounds  to  ounces. 

32.  Reduce  71  oz.  to  scruples. 

33.  Reduce  93  Ibs.  2  oz.  to  grains. 

34.  Reduce  165  oz.  to  pounds. 

35.  Reduce  962  drams  to  pounds. 

LONG  MEASURE.     (ART.  94  ) 

36.  In  636  inches,  how  many  yards  ? 
87.  In  763  feet,  how  many  rods? 


ART.  129  ]  REDUCTION'.  103 

38.  In  4  miles,  how  many  feet  ? 

39.  In  18  rods  2  feet,  how  many  inches  ? 

40.  In  1760  yards,  how  many  miles? 

41.  In  3  leagues,  2  miles,  how  many  inches? 

CLOTH   MEASURE.     (ART.  95.) 

42.  How  many  yards  in  19  quarters? 

43.  How  many  quarters  in  21  yards  and  3  quaiters? 

44.  How  many  nails  in  35  yards  and  2  quarters? 

45.  How  many  Flemish  ells  in  50  yards  ? 

46.  How  many  English  ells  in  50  yards  ? 

47.  How  many  French  ells  in  50  yards  ? 

SaUARE   MEASURE.     (ART.  96.) 

48.  In  65  sq.  yards  and  7  feet,  how  many  feet  ? 

49.  In  39  sq.  rods  and  15  yds.,  how  many  yards? 

50.  In  27  acres,  how  many  square  feet? 

51.  In  345  sq.  rods,  how  many  acres  ? 

52.  In  461  square  yards,  how  many  rods? 

53.  In  876  sq.  inches,  how  many  sq.  feet  ? 

CUBIC   MEASURE.     (ART.  97.) 

54.  In  48  cubic  yards,  how  many  feet  ? 

55.  In  54  cubic  feet,  how  many  inches  ? 

56.  In  26  cords,  how  many  cubic  feet  ? 

57.  In  4230  cubic  inches,  how  many  feet? 

58.  In  3264  cubic  feet,  how  many  cords  ? 

WINE   MEASURE.     (ART.  98.) 

59.  Reduce  94  gallons  2  qts.  to  pints. 

60.  Reduce  68  gallons  3  qts.  to  gills. 

61.  Reduce  10  hhds.  15  gallons  to  quarts. 

62.  Reduce  764  gills  to  gallons. 

63.  Reduce  948  quarts  to  hogsheads. 

64.  Reduce  896  gills  to  gallons. 


J  V4  REDUCTION.  [SECT.   IX. 

BEER  MEASURE.    (ART.  99.) 

65.  How  many  quarts  in  1 1  hogsheads  of  beer  ? 

66.  How  many  pints  in  110  gallons  2  qts.  of  beer  ? 

67.  How  many  hogsheads  in  256  gallons  of  beer? 

68.  How  many  barrels  in  320  pints  of  beer? 

69.  How  many  pints  in  46  hhds.  10  gallons  ? 

70.  How  many  hhds.  in  2592  quarts  ? 

DRY   MEASURE.     (ART.  100.) 

71.  In  156  pecks,  how  many  bushels  ? 

72.  In  238  quarts,  how  many  bushels  ? 

73.  In  360  pints,  how  many  pecks  ? 

74.  In  58  bushels,  3  pecks,  how  many  pecks  ? 

75.  In  95  pecks,  2  quarts,  how  many  quarts  ? 

76.  In  373  quarts,  how  many  bushels  ? 

77.  In  100  bushels,  2  pecks,  how  many  pints? 

TIME.    (ART.  101.) 

78.  How  many  minutes  in  16  hours  ? 

79.  How  many  seconds  in  1  day? 

80.  How  many  minutes  in  365  days  ? 

81.  How  many  days  in  96  hours  ? 

82.  How  many  days  in  3656  minutes  ? 

83.  How  many  seconds  in  1  week  ? 

84.  How  many  years  in  460  weeks? 

CIRCULAR  MEASURE.     (ART.  105.) 

85.  Reduce  23  degrees,  30  minutes  to  minutes. 

86.  Reduce  41  degrees  to  seconds. 

87.  Reduce  840  minutes  to  degrees. 

88.  Reduce  964  minutes  to  signs. 

89.  Reduce  2  signs  to  seconds. 

90.  Reduce  5  signs,  2  degrees  to  minutes. 

91.  Reduce  960  seconds  to  degrees. 

92.  Reduce  1800  minutes  to  signs. 


ART.  117.]  REDUCTION.  105 

93.  In  45  guineas,  how  many  farthings  \ 

94.  In  60  guineas,  how  many  pounds  ? 

95.  In  62564  pence,  how  many  guineas  ? 

96.  In  £84,  how  many  guineas  ? 

97.  How  many  grains  Troy,  in  46  Ibs.  7  oz.  5  pwts.  ? 

98.  How  many  pounds  Troy,  in  825630  grains  ? 

99.  Reduce  62  Ibs.  10  pwts.  to  grains. 

100.  In  16  tons,  11  cwt.  9  Ibs.,  avoir.,  how  many  pounds  ? 

101.  Reduce  782568  ounces  to  tons. 

102.  In  18  rods,  2  yds.  3  ft.  10  in.,  how  many  inches  ^ 

103.  How  many  feet  in  3  leagues,  2  miles,  12  rods  ? 

104.  In  2738  inches,  how  many  rods  ? 

105.  In  2  tons,  3  cwt.  2  qrs.  15  Ibs.,  how  many  ounces  ! 

106.  Reduce  53  Ibs.  11  pwts.  10  grs.  Troy,  to  grains. 

107.  How  many  English  ells  in  45  yards  ? 

108.  How  many  yards  in  45  English  ells  ? 

109.  How  many  Flemish  ells  in  54  yards  1 

110.  How  many  French  ells  in  60  yards  ? 

111.  In  13  m.  2  fur.  6  ft.  7  in.,  how  many  inches  ? 

112.  In  84256  feet,  how  many  leagues  ? 

113.  In  135  bu.  3  pks.  2  qts.  1  pt.  how  many  pints  ? 

114.  In  84650  pints,  how  many  quarters  ? 

115.  How  many  gills  in  48  hhds.  18  gal.  wine  measure  ? 

116.  How  many  pipes  in  98200  quarts? 

117.  How  many  seconds  in  15  solar  years  ? 

118.  How  many  weeks  in  8029200  seconds? 

119.  How  many  square  feet  in  82  acres,  36  rods,  8  yds. ! 

120.  How  many  cords  of  wood  in  68600  cubic  inches  ? 

121.  How  many  inches  in  10  cords  and  6  cubic  feet  ? 

122.  In  246  tons  of  round  timber,  how  many  inches  I 

123.  In  65200  square  yards,  how  many  acres? 

124.  In  8  signs,  43  deg.  18  sec.,  how  many  seconds  I 

125.  In  75260  minutes,  how  many  signs  ? 


106  COMPOUND    ADDITION.  [SECT.  VII. 

COMPOUND  ADDITION. 

ART.   129.    Compound  Addition  is  the  process  of 
Uniting  two  or  more  compound  numbers  in  one  sum. 

Ex.  1.  What  is  the  sum  of  £2,  3s.  4d.  1  far.;  £1,  6s. 
9d.  3  far. ;  £7,  9s.  7d.  2  far. 

Suggestion — First    write    the  Operation. 

numbers  under  each  other,  pounds  £      St     $     fart 

under  pounds,  shillings  under  shil-  2  "  4  "  4  "  1 

lings,  &c.     Then,  beginning  with  1  "  6  "  9  "  3 

H     It    Q     ff     H     ft    o 

the  lowest  denomination,  we  find  *       tf       '       * 

the  sum  is  6  farthings,  which  is  Ans.  11  '0  '9  '2 
equal  to  1  penny  and  2  far.  over.  Write  the  2  far.  under 
the  column  of  farthings,  and  carry  the  Id.  to  the  column  of 
pence.  The  sum  of  the  pence  is  21,  which  is  equal  to  Is. 
and  9d.  Place  the  9d.  under  the  column  of  pence,  and 
carry  the  Is.  to  the  column  of  shillings.  The  sum  of  the 
shillings  is  20,  which  is  equal  to  £l  and  nothing  over. 
Write  a  cipher  under  the  column  of  shillings,  and  carry  the 
£l  to  the  column  of  pounds0  The  sum  of  the  pounds  is 
11,  which  is  set  down  in  full. 

13O*  Hence,  we  derive  the  following  general 

RULE   FOB  COMPOUND   ADDITION. 

I.  Write  the  numbers  so  that  the  same  denominations 
shall  stand  under  each  other. 

II.  Beginning  at  the  right  hand,  add  each  column  sepa- 
rately, and  divide  its  sum  by  the  number  required  to  make 
ONE  of  the  next  higher  denomination.     Setting  the  remain- 
der under  tke  column  added,  carry  the  quotient  to  the  next 
column,  and  thus  proceed  as  in  Simple  Addition.    (Art.  23.) 

PROOF. — The. proof  is  the  same  as  in  Single  Addition. 

QUEST.— 121).  What  ia  Compound  Addition  1  130.  How  do  you  write  com- 
pound numbers  for  addition  1  Where  do  you  begin  to  add,  and  how  pro* 
coed  1  How  is  Compound  Addition  proved  1 


ARTS.  129,  130.]    COMPOUND  ADDITION. 


(2-) 

(3.) 

W       . 

£ 

s. 

d. 

far. 

Ib. 

oz. 

pwt.  gr. 

m. 

r. 

/*• 

^fl 

1 

3 

6 

2 

2 

5 

7 

4 

7 

15 

20 

8 

3 

0 

8 

3 

2 

0 

5 

19 

6 

4 

8 

7 

9 

18 

9 

1 

6 

8 

0 

3 

9 

6 

4 

4 

14     3     0     2^W6\11     1   13     2^/15.22  27     0 


(5.) 
£     5.    d.  /ar. 
10  17     0     1 

(6.) 
Ib.  os.  pwt.  gr. 
17  10  13     5 

(*•) 
r.  yd.  ft.  in. 

4426 

19     6     5     2 

8928 

6602 

7820 

10     4  11     3 

6814 

8263          21   11   16     6  2     3     2     S 

(8.)  (9.)  (10.) 

cwt.  qr.  Ib.  oz.  wk.  d.  hr.  mm.  yd.  qr.  na.  in. 

5345  13  4  19  30  6312 

6298     15  13  16  3231 

8172     73   5  10  7024 

6096  12  0  14  25  5112 

11.  Add  4  tons,  5  cwt.  3  qrs.  2  Ibs.  10  oz.  4  drs. ;  6 
tons,  4  cwt.  17  Ibs.  15  oz.  9  drs. ;  3  tons,  2  cwt.  1  qr.  15  Ibs. 

12.  Add  4  hhds.  10  gals.  3  qts.  1  pt. ;  15  hhds.  19  gals. 

2  qts. ;   8  hhds.  7  gals.  2  qts.  1  pt.  wine  measure. 

13.  Add  1  pipe,  1  hhd.  8  gals.  2  qts.  1  pt.  2  gills ;  1 
pipe,  6  gals.  1  qt. ;  3  pipes,  1  hhd.  3  gals.  3  qts.  1  pint. 

14.  A  man  sold  the  following  quantities  of  wheat :  5  bu. 

3  pks.  2  qts. ;  10  bu.  1  pk.  4  qts. ;  21  bu.  2  pks.  5  qts. : 
how  much  did  he  sell  in  all  ? 

15.  A  merchant  bought  3  pieces  of  silk,  one  of  which 
contained  21  yds.  2  qrs.  3  nails  ;  another  19  yds.  3  qrs.  1 
nail ;  and  the  other  26  yds.  1  qr.  and  2  nails :  how  many 
yards  did  they  all  contain  ? 


108  COMPOUND    SUBTRACTION.  [SECT.  VIIL 

COMPOUND  SUBTRACTION. 

AIIT.  131.  Compound  Subtraction  is  the  process  of 
finding  the  difference  between  two  compound  numbers. 

Ex.  1.  From  £11,  8s.  5d.  3  far.,  subtract  £5,  10s.  2d.  1 
farthing. 

Suggestion. — Write  the  less  number         Operation. 
under  the  greater,  pounds  under  pounds,      £      s.    d.far. 
shillings  under  shillings,  &c.     Then,  be-     11     8     5     3 

ginning  with  the  lowest  denomination,       5   10     2 1 

proceed  thus  :  1  far.  from  3  far.  leaves  2  5  18  3  2 
far.  Set  the  remainder  2  under  the  farthings.  Next,  2d. 
from  5d.  leave  3d.  Write  the  3  under  the  pence.  Since 
10  shillings  cannot  be  taken  from  8  shillings  ;  we  borrow 
as  many  shillings  as  it  takes  to  make  one  of  the  next 
higher  denomination,  which  is  pounds ;  and  £1,  or  20s., 
added  to  the  8s.  make  28  shillings.  Now  10s.  from  28s. 
leave  18s.,  which  we  write  under  the  shillings.  Finally, 
carrying  1  to  the  next  number  in  the  lower  line,  we  have 
£6  ;  and  £6  from  £11  leave  £5,  which  we  write  under 
the  pounds.  The  answer  is  £o,  18s.  3d.  2  far. 

132,  Hence,  we  derive  the  following  general 

RULE   FOR  COMPOUND  SUBTRACTION. 

I.  Write  the  less  number  under  the  greater,  so  that  the 
same  denominations  may  stand  under  each  other. 

II.  Beginning  at  the  right  hand,  subtract  each  lower 
number  from  the  number  above  it,  and  set  the  remainder 
under  the  number  subtracted. 

III.  When  a  number  in  the  lower  line  is  larger  than 
that  above  it,  add  as  many  units  to  the  upper  number  as  it 

UUKST.— 131.  What  is  Compound  Subtraction  ?  132.  How  do  you  write 
compound  numbers  for  subtraction  1  Where  begin  to  subtract,  and  how 
proceed?  When  a  number  in  the  lower  line  is  laiger  than  that  above  it, 
what  is  to  be  done  1  * 


ARTS.  131,  132.]  COMPOUND  SUBTRACTION.  109 

takes  to  make  ONE  c/  the  next  higher  denomination  ;  then 
subtract  as  before,  and  adding  I  to  the  next  number  in  the 
lower  line,  proceed  as  in  Simple  Subtraction. 

PROOF. — The  proof  is  the  same  as  in  Sim.  Subtraction. 

(2.)  (3.) 

From  £13,  7s.  8d.  3  far.  19  Ibs.  3  oz.  7  pwts.  12  grs. 
Take  £  6,  5s.  lid.  1  far.  15  Ibs.  8  oz.  3  pwts.  4  grs. 

(4.)  (5.) 

From  12  T.  7  cwt.  1  qr.  3  Ibs.  15  m.  3  fur.  10  r.  8  ft.  4  in. 
Take  7  T.  9  cwt.  3  qrs.  4  Ibs.  9  m.  6  fur.  3  r.  4  ft.  7  in. 

6.  From  24  yds.  2  qrs.  3  nails,  take  16  yds.  3  qrs.   2 
nails. 

7.  A  lady  having  £18,  4s.  7d.  in  her  purse,  paid  £8,  7s. 
3d.  for  a  dress  :  how  much  had  she  left  ? 

8.  If  from  a  hogshead  of  molasses  you  draw  out  19  gals. 

3  qts.  1  pi,  how  much  will  there  be  left  in  the  hogshead  ? 

9.  A  person  bought  8  tons,  3  cwt.  19  Ibs.  of  coal,  and 
having  burned  3  tons,  6  cwt.  45  Ibs.  sold  the  rest:  how- 
much  did  he  sell  ? 

10.  From  17  years,  7  mos.  16  days,  take  15  years,  and 

4  months. 

11.  From  39  yrs.  3  mos.  7  days,  4  min.,  take  23  yrs.  5 
mos.  3  days,  16  hrs. 

12.  From  43  A.  2  roods,  15  rods,  take  39  acres  and  11 
rods. 

13.  From  38  leagues,  2  miles,  5  fur.  17  rods,  take  29 
leagues,  2  miles,  7  fur.  13  rods. 

14.  From  125  bushels,  3  pecks,  4  quarts,  2  pints,  take 
108  bushels,  2  pecks,  7  quarts. 

15.  From  85  guineas,  13  shillings,  4  pence,  2  far.  take 
39  guineas,  15  shillings,  8  pence. 

QUEST,— -How  is  Compound  Subtradfcon  proved  7 


110  COMPOUND    MULTIPLICATION.        [SjECT.  VIIL 

COMPOUND  MULTIPLICATION. 

ART.  133.  Compound  Multiplication  is  the  process 
of  finding  the  amount  of  a  compound  number  repeated  or 
added  to  itself,  a  given  number  of  times. 

Ex.  1.  What  will  3  barrels  of  flour  cost,  at  £1,  7s.  5d.  2 
far.  per  barrel  ? 

Suggestion. — Write  the  multiplier  un-        ~         . 
der  the  lowest  denomination  of  the  multi-      ~  ,    1 

plicand,  and  proceed  thus  :  3  times  2  far.     i  "  V  "  5*     2  * 
are  6  far.  which  are  equal  to  Id.  and  2  3 

far.   over.      Write  the  remainder    2    far.     4 2     4     2 
under  the  denomination  multiplied,  and 
carry  the  Id.  to  the  next  product.     3  times  5d.  are  15d., 
and  1   to  carry  makes  16d.,  equal  to  Is.  and  4d.  over. 
Write  the  4d.  under  the  pence,  and  carry  the  Is.  to  the 
next  product.     3  times  7s.  are  21s.  and  1  to  carry  makes 
22s.,  equal  to  £l,  and  2s.     Write  the  2  under  the  shillings 
and  carry  the  £l  to  the  next  product.     Finally,  3  times 
£1  are  £3,  and  1  to  carry  makes  £4.     Write  the  £4  under 
the  pounds.     The  answer  is  £4,  2s.  4d.  2  far. 

134;*  Hence,  we  derive  the  following  general 

RULE  FOR  COMPOUND  MULTIPLICATION". 
Beginning  at  the  right  hand,  multiply  each  denomina- 
tion of  the  multiplicand  by  the  multiplier  separately,  and 
divide  its  product  by  the  number  required  to  make  ONE  of 
the  next  higher  denomination,  setting  down  the  remainder 
and  carrying  the  quotient  as  in  Compound  Addition. 

2.  Multiply  £4,  6s.  2d.  3  far.  by  15. 

3.  Multiply  19  Ibs.  8  oz.  9  pwts.  7  grs.  by  12. 

4.  If  a  man  walks  3  miles,  3  fur.  18  rods  in  1  hour,  how 
far  will  he  walk  in  10  hours  ? 

QUEST.— 133.  What  is  Compound  Multiplication  ?    134.  What  is  the  role 
for  Compound  Multiplication  ? 


ARTS.  133,  134.]      COMPOUND  DIVISION.  Ill 

5.  Multiply  7  leagues,  1  m.  31  rods,  12  ft.  3  in.  by  9. 

6.  Multiply  18  tons,  3  cwt.  10  Ibs.  7  oz.  3  drs.  by  11. 

7.  A  man  has  7  pastures,  each  containing  6  acres,  25 
rods,  5 1  square  feet :  how  much  do  they  all  contain  ? 

8.  A  man  bought  9  loads  of  wood,  each  containing  1 
cord  and  21  cu.  ft. :  how  much  did  they  all  contain  ? 

9.  Multiply  17  yds.  3  qrs.  2  nails  by  35. 

10.  Multiply  53  days,  19  min.  7  sec.  by  41. 

11.  Multiply  36  years,  3  weeks,  5  days,  12  hours,  by  63. 

12.  Multiply  65  hhds.  23  gals.  3  qts.  1  pt.  by  72. 

COMPOUND  DIVISION. 

135*   Compound  Division  is  the  process  of  dividing 
compound  numbers. 

Ex.  1.  A  father  divided   £10,  5s.   8d.  2  far.  equally 
among  his  3  sons  :  how  much  did  each  receive  ? 

Suggestion. — Write  the  divisor  ~ 

A,      ,  .,       ~  ,.      ,.  .,      ,  Operation. 

on  the  left  of  the  dividend,  and  x  •  ,    - 

,      .    ™     ,  TV  .  .        '  ,  £       s.      d.  far. 

proceed  as  in  Snort  Division.  Thus,  3^10  "  5  "  8      2 

3  is  contained  in  £10,  3  times  and      A    — Q  //  Q  ///>//  Ql' 

n.  ^f,  .  .  ,  ^LnS.  O         O         O         O-ar 

£1  over.     We  write  the  3  under 

the  pounds,  because  it  denotes  pounds  ;  then  reducing  the 
.remainder  £l  to  shillings  and  adding  the  given  shillings  5, 
we  have  25s.  Again,  3  is  in  25s.  8  times  and  Is.  over.  We 
set  the  8  under  the  shillings,  because  it  denotes  shillings  ; 
then  reducing  the  remainder  Is.  to  pence  and  adding  the 
given  pence  8,  we  have  20d.  Now  3  is  in  20d.  6  times 
and  2d.  over.  We  set  the  6d.  under  the  pence,  because  it 
denotespence.  Finally,  reducing  the  rem.  2d.  to  farthings 
and  adding  the  given  far.  2,  we  have  10  far. ;  and  3  is  in 
10,  3  times  and  1  far.  over.  Write  the  3  under  the  far 

QUEST.— 135.  What  is  Compound  Division  2 


112  COMPOUND  DIVISION.  [SECT.  VIIL 

136.  Hence,  we  derive  the  following  general 

KULE   FOR   COMPOUND   DIVISION. 

1.  Beginning  at  the  left  hand,  divide  each  denomination 
of  the  dividend  by  the  divisor,  and  write  the  quotient  fig- 
ures under  the  figures  divided. 

II.  If  there  is  a  remainder,  reduce  it  to  the  next  lower 
denomination,  and  adding  it  to  the  figures  of  the  correspond- 
ing denomination  of  the  dividend,  divide  this  number  as 
before.  Thus  proceed  through  all  the  denominations,  and 
the  several  quotients  will  be  the  answer  required. 

OBS.  1.  Each  quotient  figure  is  of  the  same  denomination  as  that 
part  of  the  dividend  from  whitfia  it  arose. 

2.  When  the  divisor  exceeds  12,  and  is  a  composite  number,  we 
may  divide  first  by  one  factor  and  that  quotient  by  the  other. 

2.  Divide  14  Ibs.  5  oz.  6  pwts.  9  grs.  by  3. 

3.  Divide  £5,  17s.  8cl.  1  far.  by  4. 

4.  Divide  25  Ibs.  3  ounces,  8  pwts.  7  grs.  by  5. 

5.  Divide  15  T.  15  cwt.  3  qrs.  10  Ibs.  by  6, 

6.  Divide  23  yards,  2  qrs.  1  nail,  by  7. 

7.  Divide  35  leagues,  1  rn.  3  fur.  17  rods  by  8. 

8.  Divide  45  hhds.  18  gals.  39  qts.  1  pint  by  9.  » 

9.  A  farmer  had  34  bu.  3  pks.  1  qt.  of  wheat  in  9  bags . 
how  much  was  in  each  bag? 

10.  If  you  pay  £25,  17s.  8^d.  for  5   cows,  how  much 
will  that  be  apiece  ? 

11.  Divide  38  tons,  5  cwt.  2  qrs.  15  Ibs.  by  17. 

12.  Divide  41  hhds.  13  gals.  2  qt.  wine  measure  by  23 

13.  Divide  54  acres,  2  roods,  25  rods,  by  34. 

14.  Divide  29  cords,  19  cu.  feet,  18  cu.  inches  by  41. 

15.  Divide  78  years,  17  weeks,  24  days,  by  63. 


QUEST.— 136.  What  is  the  rule  for  Compound  Division  ?    Obs.  Of  what  de- 
nomination is  each  quotient  figure  ? 


MISCELLANEOUS    EXERCISES.  118 

MISCELLANEOUS   EXERCISES. 

1.  From  the  sum  of  463279 +  '734658,  take  926380. 

2.  To  the  difference  of  856273  and  4671 9,  add  420376. 

3.  To  476208  add  5207568  —  4808345. 

4   Multiply  the  sum  of  863576  +  435076  by  287. 

5.  Multiply  the  difference  of  870358  —  640879  by  365. 

6.  Divide  the  sum  of  439409  +  87646  by  219. 

7.  Divide  the  difference  of  607840  — 23084  by  367 

8.  Divide  the  product  of  865060X406  by  1428. 

9.  Divide  the  quotient  of  55296+144  by  89. 

10.  What  is  thesum  of  4845  +  76  +  1009  +  463+407  ? 

11.  What  is  the  sum  of  836X46,  and  784x76? 

12.  What  is  the  sum  of  1728+72,  and  2828-+ 96? 

13.  What  is  the  sum  of  85263  —  45017,  and  68086? 

14.  What  is  the  difference  between  38076  +  16325,  and 
20268  +  45675? 

15.  What  is  the  difference  between  40719  +  6289,  and 
31670  —  18273. 

16.  What  is  the  difference  between  378X  96,  and  9419  I 

17.  What  is  the  difference  between  7560-7-504,  and 
7560X504? 

18.  Froml45X87,  take  12702+87. 

19.  Multiply  83X19  by  75X23. 

20.  How  many  times  can  34   be  subtracted  from  578  f 

21.  How  many  times  can  1512  be  taken  from  7569  ? 

22.  How  many  times  can  63  X  24  be  taken  from  27640 1 

23.  How  many  times  is  68  +  31  contained  in  45600? 

24.  Divide  832  +  1429  by  45  +  84. 

25.  Divide  467  +  2480  by  346  —  187. 

26.  Divide  68240—16226  by  10405  —  6200. 
27    Divide  320X160  by  2125  —  960. 

28.  Divide  826340  —  36585  by  126X84. 

29.  From  62345  +  19008,  take  2134X38. 

30.  From  2631X216,  take  576—36. 


1  14  MISCELLANEOUS    EXERCISES. 

33.  A  young  man  having  50  dollars,  bought  a  coat  frf 
15  dollars,  a  pair  of  pants  for  8  dollars,  a  vest  for  5  dol- 
lars, and  a  hat  for  3  dollars :  how  much  money  did  he 
have  left  ? 

34.  A  farmer  sold  a  cow  for  18  dollars,  a  calf  ror  4 
dollars,  and  a  lot  of  sheep  for  35  dollars:  how  much 
more  did  he  receive  for  his  sheep  than  for  his  cow  and 
calf? 

35.  A  man  having  90  dollars  in  his  pocket,  paid  27 
dollars  for  9  cords  of  wood,  35  dollars  for  7  tons  of  coal, 
and  1 1  dollars  for  carting  both  home :  how  much  money 
had  he  left  ? 

36.  A  young  lady  having  received  a  birthday  present 
of  100  dollars,  spent  17  dollars  for  a  silk  dress,  26  dol- 
lars for  a  crape  shawl,  and  8  dollars  for  a  bonnet :  how 
many  dollars  did  she  have  left  ? 

37.  A  dairy- woman  sold  23  pounds  of  butter  to  one 
customer,  34  pounds  to  another,  and  had  29  pounds  left: 
how  many  pounds  had  she  in  all  ? 

38.  A  lad  bought  a  pair  of  boots  for  16  shillings,  a 
pair  of  skates  for  10  shillings,  a  cap  for  17  shillings,  and 
had  20  shillings  left :  how  many  shillings  had  he  at  first  ? 

39.  A  grocer  having  500  pounds  of  lard,  sold  3  kegs 
of  it ;  the  first  keg  contained  43  pounds,  the  second  45 
pounds,  and  the  third  56  pounds  :  how  many  pounds  did 
he  have  left  ? 

40.  A  man  bought  a  horse  for  95  dollars,  a  harness 
for  34  dollars,  and  a  wagon  for  68  dollars,  and  sold  them 
all  for  225  dollars  :  how  much  did  he  make  by  his  bar- 
gain? 

41.  A  person  being  1000  miles  from  home,  on  his  re- 
turn, traveled  150  miles  the  first  day,  240  miles  the  sec- 
ond day,  and  310  miles  the  tfiird  day:  how  far  from 
home  was  he  then  ? 


MISCELLANEOUS    EXERCISES.  115 

\  42.  George  bought  a  pony  for  78  dollars  and  paid  3 
dollars  for  shoeing  him ;  he  then  sold  him  for  100  dol- 
lar :  how  much  did  he  make  by  his  bargain  ? 

43.  A  man  bought  a  carriage  for  273  dollars,  and  paid 
27  dollars  for  repairing  it ;  he  then  sold  it  for  318  dol- 
lars :  how  much  did  he  make  by  his  bargain  ? 

44.  A  man  bought  a  lot  for  275  dollars,  and  paid  a 
carpenter  850  dollars  for  building  a  house  upon  it :  he 
then  sold  the  house  and  lot  for  1200  dollars :  how  much 
did  he  make  by  the  operation  ? 

45.  A  farmer  having  150  sheep,  lost  17  and  sold  65 ; 
he  afterwards  bought  38  :  how  many  sheep  had  he  then  ? 

46.  A  man  bought  27  cows,  at  31  dollars  per  head: 
how  many  dollars  did  they  all  cost  him  ? 

47.  A  miller  sold  251  barrels  of  flour,  at  8  dollars  ^ 
barrel :  how  much  did  it  come  to  ? 

48.  A  merchant  sold  218  yards  of  cloth,  at  8  dollars 
per  yard :  how  much  did  it  come  to  ? 

49.  A  merchant  sold  18  yards  of  broadcloth,  at  4  dol- 
lars a  yard,  and  21  yards  of  cassimere,  at  2  dollars  a  yard : 
how  much  did  he  receive  for  both  ? 

50.  A  farmer  sold  12  calves,  at  5  dollars  apiece,  and 
35  sheep,  at  3  dollars  apiece :  how  much  did  he  receive 
for  both  ? 

51.  A  grocer  sold  to  one  person  25  firkins  of  butter, 
at  7  dollars  a  firkin,  and  13  to  another,  at  8  dollars  a  fir- 
kin :  how  much  did  both  lots  of  butter  come  to  ? 

52.  A  shoe  dealer  sold  100  pair  of  coarse  boots  to  one 
customer,  at  4  dollars  a  pair,  and  156  pair  of  fine  boots 
to  another,  at  5  dollars  a  pair:  what  did  both  lots  of 
boots  come  to  ? 

53.  A  miller  bought  165  bushels  of  corn,  at  5  shillings 
a  bushel,  and  286  bushels  of  wheat,  at  9  shilliigs  a 
bushel :  how  much  did  he  pay  for  both  ? 


.  16  MISCELLANEOUS    EXERCISES. 

54.  A  man  bought  45  clocks,  at  3  dollars  apiece,  and 
sold  them,  at  5  dollars  apiece :  how  much  did  he  make  by 
his  bargain? 

55.  A  bookseller  bought  87  books,  at  7  shillings  apiece, 
and  afterwards  sold  them,  at  6  shillings  apiece :  how  much 
did  he  lose  by  the  operation? 

56.  How  many  yards  of  calico,  at  18  cents  a  yard,  can 
be  bought  for  240  cents  ? 

57.  A  little  girl  having  326  cents,  laid  it  out  in  ribbon, 
at  25  cents  a  yard :  how  many  yards  did  she  buy  ? 

58.  If  a  man  has  500  dollars,  how  many  acres  of  land 
can  he  buy,  at  15  dollars  per  acre? 

59.  How  many  cows,  at  27  dollars  apiece,  can  be  bought 
for  540  dollars  ? 

60.  How  many  barrels  of  sugar,  at  23  dollars  per  bar- 
rel, can  a  grocer  buy  for  575  dollars? 

61.  Henry  sold  his  skates  for  87  cents,  and  agreed  to 
take  his  pay  in  oranges,  at  3  cents  apiece :    how  many 
oranges  did  he  receive  ? 

62.  William  sold  80  lemons,  at  4  cents  apiece,  and  took 
his  pay  in  chestnuts,  at  5  cents  a  quart :  how  many  chest- 
nuts did  he  get  for  his  lemons  ? 

63.  A  milkman  sold  110  quarts  of  milk,  at  6  cents  a 
quart,  and  agreed  to  take  his  pay  in  maple  sugar,  at  11 
cents  a  pound  :  how  many  pounds  did  he  receive  ? 

64.  A  farmer  bought  25  yards  of  cloth,  which  was 
worth  6  dollars  per  yard,  and  paid  for  it  in  wood,  at  2 
dollars  per  cord :  how  many  cords  did  it  take  ? 

65.  A  pedlar  bought  4£  pieces  of  silk,  at  24  dollars 
apiece :  how  much  did  he  pay  for  the  whole  ? 

66.  A  farmer  sold  8-j-  bushels  of  wheat,  at  96  cents 
per  bushel :  how  much  did  he  receive  for  his  wheat  ? 

67.  A  man  sold  a  lot  of  land  containing  15f  acres,  at 
16  dollars  per  acre :  how  much  did  he  receive  for  it  ? 


MISCELLANEOUS    EXERCISES.  117 

\  68.  If  a  man  can  walk  45  miles  in  a  day,  how  far  caa 
ha  wak  in  2  Of  days? 

69.  What  cost  75  yds.  of  tape,  at  f  of  a  cent  per  yd.  ? 

70.  What  will  100  pair  of  childrens'  gloves  come  to, 
at  -ft  of  a  dollar  a  pair  ? 

71.  What  will  160  boys'  caps  cost,  at  f  of  a  dollar 
apiece  ? 

72.  What  will  210  pair  of  shoes  cost,  at  -f-  of  a  dollar 
a  pair  ? 

73.  How  many  childrens'  dresses  can  be  made  from  a 
piece  of  lawn  which  contains  54  yards,  if  it  takes  4£  yards 
for  a  dress  ? 

74.  A  farmer  wishes  to  pack   100  dozen  of  eggs  in 
boxes,  and  to  have  each  box  contain  6-J-  dozen :  how  many 
boxes  will  he  need  ? 

75.  A  lad  having  275  cents,  wishes  to  know  how  many 
miles  he  can  ride  in  the  Railroad  cars,  at  2£  cents  per  mile : 
how  many  miles  can  he  ride  ? 

76.  How  many  apples,  at  £  a  cent  apiece,  can  Horatio 
buy  for  75  cents  ? 

77.  If  Joseph  has  to  pay  f  of  a  cent  apiece  for  marbles, 
how  many  can  he  buy  for  84  cents  ? 

78.  At  f  of  a  dollar  apiece,  how  many  parasols  can  a 
.shopkeeper  buy  for  168  dollars? 

79.  If  I  am  charged  -f-  of  a  dollar  apiece  for  fans,  how 
many  can  I  buy  for  265  dollars  ? 

80.  How  many  yards  of  silk,  which  is  worth  ^  of  a 
dollar  a  yard,  can  I  buy  for  227  dollars  ? 

81.  How  many  pair  of  slippers,  at  -J  of  a  dollar  a  pair, 
can  be  bought  for  448  dollars  ? 

82.  In  £45, 13s.  6d.,  how  many  pence  ? 

83.  In  £63,  7s.  8d.  2  far.,  how  many  farthings? 

84.  How  many  yards  of  satin  can  I  buy  for  £75, 10s., 
If  I  have  to  pay  5  shillings  per  yard  ? 


118  MISCELLANEOUS    EXERCISES. 

85.  How  many  six-pences  are  there  in  £100  ? 

86.  A  grocer  sold  10  hogsheads  of  molasses,  at  3  shit 
lings  per  gallon :  how  many  shillings  did  it  come  to  ? 

87.  A  milkman  sold  125  gallons  of  milk,  at  4  cents  pel 
quart :  how  much  did  he  receive  for  it  ? 

88.  A  man  made  30  barrels  of  cider  which  he  wished  to 
put  into  pint  bottles :  how  many  bottles  would  it  require  ? 

89.  How  much  would  85  bushels  of  apples  cost,  at  12 
cents  a  peck  ? 

90.  What  will  97  pounds  of  snuff  cost,  at  8  cents  per 
ounce  ? 

91.  What  will  5  tons  of  maple  sugar  come  to,  at  11 
cents  a  pound  ? 

92.  A  farmer  sold  34  tons  of  hay,  at  65  cents  per  hun- 
dred :  how  much  did  he  receive  for  it  ? 

93.  A  blacksmith  bought  53  tons  of  iron  for  3  dollars 
per  hundred :  how  much  did  he  pay  for  it  ? 

94.  A  young  man  returned  from  California  with  50 
pounds  of  gold  dust,  which  he  sold  for  16  dollars  per 
ounce  Troy :  how  much  did  he  receive  for  it  ? 

95.  A  man  bought  36  acres  of  land  for  3  dollars  per 
square  rod :  how  much  did  his  land  cost  him  ? 

96.  John  Jacob  As  tor  sold  five  building  lots  in  the  city 
of  New  York,  containing  560  square  rods,  for  13  dollars 
per  square  foot :  how  much  did  he  receive  for  them  ? 

97.  A  laboring  man  engaged  to  work  5  years  for  16 
dollars  per  month :  what  was  the  amount  of  his  wages  ? 

98.  What  will  17  cords  of  wood  cost,  at  6  cents  per 
cubic  foot  ? 

99.  If  it  takes  35  men  18  months  to  build  a  fort,  how 
many  years  would  it  take  1  man  to  build  it  ? 

]  00.  If  it  takes  1  man  360  days  to  build  a  house,  how 
many  weeks  would  it  take  15  men  to  build  it,  allowing  6 
working  days  to  a  week  ? 


ANSWERS  TO  EXAMPLES. 


119 


ANSWERS  TO  EXAMPLES. 
ADDITION. 


Sx.              Aus. 

Ex.               Ana. 

Ex.               Ans. 

ART.  20. 

4.  5286  yards. 

28.  171658. 

1.  Given. 

5.  2404. 

29.  57  dollars. 

2.  68. 

6.  2765. 

30.  58  dollars. 

3.  589. 

7.  10040. 

31.  120  dollars. 

4.  768. 

8.  8668. 

32.  565. 

5.  9987. 

9.  84  inches. 

33.  742. 

6.  878. 

10.  114  feet 

34.   1530. 

V.  6767. 

11.  168  dollars. 

35.  1779. 

8.  8898. 

12.  192  rods. 

36.  1597. 

9.  8779. 

13.  782  pounds. 

37.  1757. 

10.  6796. 

14.   1380  yards. 

38.  2379. 

11.  88776. 

15.  576  miles. 

39.  2619. 

12.  986788. 

16.  836  sheep. 

40.  1020. 

17.  615  dollars. 

41.   1418. 

ART.  22. 

18.   181  dollars. 

42.  1191. 

13,  14.  Given. 

19.   1452. 

43.  150  bushels. 

15.   1454. 

20.   1255. 

44.   133  yards. 

16.  15300. 

21.   1881. 

45.  731  acres. 

17.  13285. 

22.  6693. 

46.  1197  cattle. 

23.  20485. 

47.  12554  dollars. 

ART.  24. 

24.  9726. 

48.  1282. 

1.  155  pounds. 

25.  1769. 

49.  2528. 

2.  413  feet. 

26.   1500. 

50.  365  days. 

3.  1960  dollars. 

27.  106284. 

ART.  2  4.  a. 

10.  65471. 

20.  551452. 

30.  279,075. 

1.  300. 

11.  327371. 

21.  46157. 

31.  295,306. 

2.  6000. 

12.  390497. 

22.  424634. 

32.  1,606,895. 

3.  9000. 

13.  37938. 

23.  430032. 

35.  6,140,704. 

4.  4861. 

14.  50342. 

24.  3458772. 

36.  7,569,904. 

5.  4871. 

15.  449458. 

25.  48350. 

37.  9,253,854. 

6.  47067. 

16.  466789. 

26.  514299. 

38.  9,247,176. 

7.  53340. 

17.  40290. 

27.  595522. 

39.  10,531,960 

8.  59139. 

18.  50676. 

28.  5781566. 

40.  12,811,860. 

9.  61304. 

19.  508302. 

29.  61993. 

120 


ANSWERS.          [PAGES  28 — 35. 


SUBTRACTION. 


Ex.               A.ns. 

Ex.               Ans. 

Ex.              Ana. 

ART.  28. 

14.  275  pounds. 

48.  222  bushels. 

1.  Given. 

15.  613  yards. 

49.  195  dollars. 

2.  24. 

16.  310  rods. 

50.  1122  dollars. 

3.  12. 

17.  230  gallons. 

51.  1659  dollars. 

4.   153. 

18.  503  hhds. 

52.  3023  dollars. 

5.  24S. 

19.  76  bushels. 

53.  1763  dollars. 

6.  31  dollars. 

20.  127  dollars. 

54.  3747  dollars. 

7.  12  pounds. 

21.  249  pounds. 

55.  16014  dollars. 

8.  115  yards. 

22.   1082  rods. 

56.   1315  dollars. 

9.  222  shillings. 

23.   13016. 

57.  5385  dollars. 

10.  222  marbles. 

24.  310768. 

58.  5735  dollars. 

25.  464374. 

59.  13944  soldiers 

ART.  3O. 

26.  5244038. 

60.  94760000  m. 

11,  12.  Given. 

27.  45. 

61.  17  oranges. 

13.  137. 

28.  308. 

62.  33  marbles. 

14.  2616. 

29.  240. 

63.  76  sheep. 

15.  3270. 

30.  58. 

64.  52  cents. 

16.  3203. 

31.  542. 

65.  43  yards. 

17.  5365667. 

32.  2021. 

66.  122  dollars. 

33.  1825. 

67.  87  dollars. 

ART.  32. 

34.  2600. 

68.  66  pears. 

1.  217.* 

35.  3085. 

69.  59. 

2.  182. 

36.   1306. 

70.  164. 

3.  242. 

37.  4098. 

71.  149  pounds. 

4.  369. 

38.  1108. 

72.  164  bushels. 

5.  1029. 

39.  4531. 

73.  263  miles. 

6.  1008. 

40.  14520. 

74.  125  gallons. 

7.  3289. 

41.  24622. 

75.  179  pounds. 

8.  3434. 

42.   125028. 

76.  175  dollars. 

9.  35100. 

43.  64303. 

77.  339  pounds. 

10.  312657. 

44.  224066. 

78.  172  barrels. 

11.  1. 

45.   103875. 

79.  297  pages. 

12.  23  dollars. 

46.  420486. 

80.  110  dollars. 

13.  57  bushels. 

47.  72  sheep. 

81.  392  dollars. 

*  It  is  an  excellent  exercise  for  the  pupil  to  prove  all  the  examples.    This  is 
one  of  the  beet  means  to  give  him  confidence  in  his  own  powers. 


PAGES  39 — 46  ]  ANSWERS. 


121 


MULTIPLICATION. 


Ex.               Ans. 

Ex.              Ans. 

Ex.              Ana. 

ART.  39. 

ART.  41. 

33.  9100  weeks. 

1.  Given, 

34  —  37.  Given. 

34.  23760  min. 

2.   68. 

35.  28350  gallons. 

3.  936. 

ART.  43. 

36.  34675  "dolls. 

4.  8084. 

1.  252. 

37.  33840  sq.  in. 

5.  5550. 

2.  390. 

38.  26070  miles. 

6.   12066. 

3.  567. 

7.  24408. 

4.  582. 

ART.  45. 

8.   35550. 

5.  840. 

40.  Given. 

9.  56707. 

6.  1155. 

41.  260. 

10.  Given. 

7.  3568. 

42.  3700. 

8.  2763. 

43.  51000. 

ART.  40. 

9.  3920. 

44.  226000. 

11.  312. 

10.  460. 

45.  341000. 

12.  480. 

11.  572. 

46.  46900*00. 

13.  249. 

12.  816. 

47.  52300000. 

14.  840. 

13.   1092. 

48.  681000000. 

15.  828. 

14.  1170. 

49.  856120000. 

16.  815. 

15.  2185. 

50.  96030500000 

17.   2248. 

16.  4515. 

51.  Given. 

18.   3144. 

17.   12306. 

19.  2520. 

18.  25355. 

ART.  46. 

20.   1900. 

19.  342  dollars. 

52.   17000. 

21.  3960. 

20.  336  bushels. 

53.  291000. 

22.   656C, 

21.  336  inches. 

54.  4920000. 

23.  5628. 

22.  620  pounds. 

55.   11700000 

24.  8712. 

23.  391  dollars. 

56.  33930. 

25.   1050  dollars. 

24.  475  dollars. 

57.  789600. 

26.  2300  dollars. 

25.  1591  dollars. 

58.   16170000. 

27.   1372  dollars. 

26    1950  shil. 

59.   262660000. 

28,  2720  dollars. 

27.  1575  dollars. 

60.  7oOO  minutes. 

29.  4837  dollars. 

28.  2430  shil. 

61.  2400  dollars. 

30,  7785  dollars. 

29.  3936  ounces. 

62.  6800  shillings. 

31.  7744  dollars. 

30.  10754  dollars. 

63    27000  dollars. 

S2.  8820  dollars. 

31.  6710  miles. 

64.  352500  days. 

&3.  2  1285  dollars. 

32.   8760  hours. 

122 


ANSWERS. 


[PAGES  47 — 55. 


MULTIPLICATION    CONTINUED. ARTS.  47,  48. 


Ex.     Ana. 

Ex.       Ans.        Ex.       Ans. 

65.  Given. 

78.  2520000.    [  91.  5816049  galls. 

66.  19500. 

79.  65000000.     92.  101198340  d. 

67.  40800. 

80.  722000000.    93.  146460440  T. 

68.  504000. 

81.  21000000000.  94.  1190439180. 

69.  800000. 

82.  72800000000.  95.  3759670728. 

70.  3300000. 

83.  2240000yds.   96.  4223213600. 

71.  14620000. 

84.  140000  miles.  97.  5815178600. 

72.  65360000. 

85.  700000  dolls.   98.  12976172335. 

73.  104520000. 

86.  504000  dolls.   99.  124811441568 

74.  183244000. 

87.  27375000  d.  100.  54719418834. 

75.  Given. 

88.  367608  Ibs.   101.  469234745451 

76.  420000. 

89.  3838460ft.   102.  197118900. 

77.  442000. 

90.  4217202  r.    103.  420152303451, 

SHORT  DIVISION. 

ART.  54. 

17.  25. 

9.  116*. 

1.  Given. 

18.  76. 

10.  728. 

2.  21. 

19.  456. 

11.  1552f. 

3.  23. 

12.  1004f 

4.  122. 

ART.  57. 

13.  400f. 

5.  111. 

20.  Given. 

14.  903*. 

6.  342. 

21.  509. 

15.  923. 

7.  1122. 

22.  901. 

16.  1222f. 

8.  1321. 

23.  1067. 

17.  875. 

9.  1111. 

24.  503. 

18.  1011-|. 

25.  Given. 

19.  63  pair. 

ART.  55. 

20.  42  hats. 

10.  Given. 

ART.  61. 

21.  24  marbles. 

11.  71. 

1.  142. 

22.  45  children. 

12.  43. 

2.  101-J-. 

23.  75  yards. 

13.  412. 

3.  76. 

24.  85  barrels,  an  C 

14.  411. 

4.  75. 

5  dolls,  over. 

5.  102f 

25.  92  days. 

ART.  56. 

6.  56|. 

26.  158-J-  yards. 

15.  Given. 

7.  120f. 

27.  195  hours. 

16.  14. 

8.  95. 

28.  333|  hours 

PAGES  56 62.]  ANSWERS. 


123 


LONG  DIVISION. 


far 


Ex. 


Ex. 


ART.  62. 

1,  2.  Given. 

3.  128.* 

4.  364. 

5.  1825f. 

6.  533. 

7.  732. 

8.  931. 
9—11.  Given. 

ART.  65. 

1.  46-i 

2.  48-f. 

3.  80f. 

4.  40^. 

5.  58-ft. 

6.  48. 
7. 

8. 

9.  41A- 

10.  27. 

11.  23f£. 

12.  21-fj, 

13.  19ff. 

14.  20. 
15. 

16, 

17.  45f|. 

18.  57ff. 

19.  24  caps. 

20.  35  pair. 

21.  28  barrels. 


82.  1900W. 

83.  840f£f. 

QA  g>7  \   5  0  5 

OT:.  O  t  ^r~i   ,;'  .>T, 


1 22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 

41. 
42. 

43. 

44. 
45. 
46. 

47. 
48. 
49. 
50. 

85. 
86. 

87. 

88. 


16-ft- shillings 
10^  pounds. 
16|f  pounds. 
17  trunks. 
30  weeks. 
32f£  yards. 
75  dresses. 
81  sheep. 
73-J-f-  acres. 
61  shares. 
3 Iff  years. 
48ii  hhds. 
43ff-  months. 
5  Iff  months. 
50  dollars. 
lOif  months. 
90  pounds. 
60,  and  1  over. 
106,  and  22 

over. 
26,    and    28 

over. 
42,    and    28 

over. 
30|f. 
34. 
53^. 


25f£. 


51. 
52. 

53.  218-ftV 

54.  216-iftfr. 

ART.  67. 

55.  56.  Given. 

57. 
58. 
59. 
60. 
61. 
62. 
63. 


1620-fiif 


ART.  68. 

64,  65.  Given, 

66. 

67. 

68. 

69. 

70.  46<HHH!. 

71. 

72. 

73. 

74.  27«rH. 

75. 

76. 

77. 

78. 

79.  30. 

80.  14834fJ-. 
89. 

90. 
91. 


124 


ANSWERS. 


[PAGES  68  -92, 


FRACTIONS. 


Ex,           Ans. 

Ex.           Ana. 

Ex.           Ans. 

Ex.           Ans. 

ART.  83. 

15.    147. 

30.  1896. 

11.  307£. 

3.  10  doll. 

16.    135. 

31.  Given. 

12.  273. 

4.  10  shill. 

19.  70. 

32.  1122  r. 

13.  304. 

5.  7f  doll. 

20.  117. 

33.  752|  dol. 

14.  329+. 

6.  21  shill. 

21.   189. 

34.   1609fd. 

ART.  86. 

7.  36. 

22.   217. 

35.  6120  dol. 

3.   12. 

8.  28. 

23.   112. 

ART.  85. 

4.  6. 

9.  6. 

24.   399. 

5.   100. 

5.  4-fr 

10.  14. 

25.   200. 

6.   190. 

6.  5ff. 

11.  40. 

26.  270. 

7.  423. 

V.  5«. 

12.  21. 

27.  575. 

8.   1260. 

8.  11H. 

13.  32. 

28.   1287. 

9.   108. 

9.  9/ff. 

14.  100. 

29.   1540. 

10.   118f. 

10.  9«. 

EXAMPLES  FOR  PRACTICE. 


1.  8  a. 

8.  9|  bus. 

15.  25  W  a. 

22.  277-f  1. 

2.  8  p. 

9.  I7f  yds. 

16.  28  W  bar. 

23.  16  bar. 

3.  24  yds. 

10.  30  lambs. 

17.  48  yds. 

24.  20  hrs. 

4.  13|  yds. 

11.  16  rods. 

18.  22  miles. 

25.  10  bar. 

5.   lOlbs. 

12.   8  rods. 

19.  20-H-  cts. 

26.  24  colts. 

6.  24  yds. 

13.  1  7-2^  yds. 

20.   12  mo. 

27.  32  wag. 

7.  49  mar. 

14.  10  cows. 

21.  24  cattle. 

ADDITION  OF  FEDERAL  MONEY.     ART.  117. 

2.^1278.699. 

6.  $1743.828. 

10.  $978.297. 

14.$829.49d 

3.111261.52. 

7.12478.735. 

11.  $2037.379. 

15.  $34.75. 

4.  $2622.337. 

8.110224.78. 

12.  $880.317. 

16.  $74.375. 

5.85599.332. 

9.  $12858.266, 

13.  $301.243. 

17.  $162.06. 

SUBTRACTION  OF  FEDERAL  MONEY.      ART.  118. 

2.  $468.851. 

6.  $877.155.  1 

0.  $81980.755. 

14.  $49.928. 

3.  $497.73. 

7.  $461.543.  ] 

il.  $67671.  133. 

15.  $357.04. 

4.  $527.247. 

8.  $435.103.  1 

L2.  $0.89. 

16.  $2.125. 

5.  §5916.707. 

9.  $1461.78.1 

13.  $2.317. 

17.  $1.945. 

PAGES  93 — 100.]  ANSWERS.  125 

MULTIPLICATION  OF  FEDERAL  MONEY.    ART.  119. 


Ex.           Ana, 

Ex.            Aria. 

Ex.         Ana. 

Ex.         Ans. 

1.  Given. 

2.  $5070. 
3.  $7250.625. 
4,  $21097.  80. 
5.  $335636.62 
6.  $255991.68 

7.  $458122.602 
8.  $773262.87. 
9.$2182139.52 
10.  $1.36. 
11.  $10.44. 
12.  $31. 

13.  $78.75. 
14.$12.375. 
15.  $39.45. 
16.  $9.375. 
17.  $23.75. 
18.11181.28 

19.  $3346.50 
20.  $1495. 
21.  $4238.08 
22.  $7.50. 
23.  $73.50. 
24.  $279.50. 

DIVISION  OF  FEDERAL  MONEY.     ART.  120. 


1.  Given. 

7.  $9933.57. 

13.  $8902.627. 

19.  $26.82 

2.  $142.712. 

8.  $11.322. 

14.  $972.38. 

20.  $8.35. 

3.  $1195.956. 

9.  $110.57. 

15.  $40.69. 

21.  $2.767. 

4.  $806.012. 

10.  $68.47. 

16.  $6.12. 

22.  $1.738. 

5.  $32.16. 

11.  $92.09. 

17.  $7.31. 

23.  $6.807. 

6.  $96.70. 

12.  $49.32. 

18.  $20.16. 

REDUCTION  DESCENDING. 

ART.  124. 
1-4.  Given. 
5.  4320d. 
6.  469s. 

11.  7348  gr. 
13.  212  oz. 
14.  67  Ibs. 
15.  1728  dr. 

20.  24640  r. 
21.  56  qrs. 
22.  344  na. 
23.  286  na. 

28.  180  pk. 
29.  52  pts. 
30.  1680  qts. 
31.  2520  hrs. 

7.  827  far. 
8.  Given. 
9.  156  oz. 
10.  1020pwt. 

16.  19696  oz. 
17.  5120  r. 
18.  5568  in. 
19.  12614  ft. 

24.  92  qts. 
25.  976  g. 
26.  2016  g. 
27.  10332  q. 

32.  36000  m. 
33.  954000s. 
34.  524160  m. 
35.  5875200s 

REDUCTION  ASCENDING. 


ART.  12T. 
1—4.  Given. 
5.  27  shillings. 
6.  20  shillings. 
7.  £l,6s.0d.2far. 
8.  22  Ibs.  1  oz. 
9.    3  Ibs.  0   oz.   8 
pwts. 

10.  1    oz.   2  pwts. 
20  grs. 
11.  39  Ibs.  14  oz. 
12.  29  qrs.  11  Ibs. 
13.  1    cwt.   4    Ibs. 
11  oz. 
14.  3  Ibs.  14  oz.  8 
drs. 

15.  2  tons. 
16.  24  >ds.  1  in. 
17.  60  r.  10  ft. 
18.  8  miles. 
19.  1  m.  6  fur.    32 
r.  5  yds. 
20.  2   lea.  1    m.  3 
fur.  5  r. 

126 


ANSWERS.          LPAaE*  101 — 1 


REDUCTION   ASJENDING  AND   DESCENDING. 


Ex.              Ans. 

Ex.               Ans» 

Ex.               Ans. 

ART.  129. 

31.  540  ounces. 

62.  23  gals.  3  qts. 

1.  316  cents. 

32.  1704  scruples. 

Ipt. 

2.  812  mills. 

33.  536640  grs 

63.  3hhds.  48  gls, 

3.  2  dolls.  45  cts. 

34.   13  Ibs.  9  oz. 

64.   28  gals. 

4.  3  dimes  2  cts. 

35.   10  Ibs.    0  oz. 

65.  2376  qts. 

1  mill. 

2  drs. 

66.  884  pints. 

5.  95000  cents. 

36.  17  yds.  2  ft. 

67.  4hhds.40gls. 

6.  16000  cents. 

37.  46  rods  4  ft. 

68.   1  bbl.  4  gals. 

7.  3170  dimes. 

38.  21120  feet. 

69.   19952  pts. 

8.  4  dolls.  56  cts. 

39.  3588  inches. 

70.   12  hhds. 

1  mill. 

40.   1  mile. 

71.   39  bushels. 

9.  8E.2dolls.50c. 

41.  696960  in. 

72.  7   bu.    1    pk 

10.  61123  mills. 

42.  4  yds.  3  qrs. 

6  qts. 

11.  356  shillings. 

43.  87  qrs. 

73.  22  pks.  4  qts. 

12.  938  farthings. 

44.  568  nails. 

74.  235  pecks. 

13.  £5. 

45.  66Fl.e.2qrs. 

75.  762  quarts. 

14.   75s.  6d.  2  far. 

46.  40  E.  e. 

76.   11  bu.  2  pks 

15.   18240  far. 

47.  33  F.  e.  2  qrs. 

5  qts. 

16.  60  shillings. 

48.   592  sq.  ft. 

77.  6432  pints. 

17.  5082  pence. 

49.   1194f  sq.  yds. 

78.  960  minutes. 

18.  £3,    17s    6d. 

50.   H76120sq.ft. 

79.   86400  sec. 

1  far. 

51.   2  A.  25  sq.r. 

80.  525600  min. 

19.  84  ounces. 

52.   15    sq.    r.    7f 

81.  4  days. 

20.  2:00  pwts. 

sq.  yds. 

82.   2  days  12  hr? 

21.   13  Ibs.  9  oz. 

53.  6    sq.    ft.     12 

56  min. 

22.   1  oz.  15  pwts. 

sq.  in. 

83.  604800  sec, 

23.   19735  grains. 

54.  1296  cu.  ft. 

84.  8  yrs.  11  mo. 

24.   1  Ib.  2  oz.    5 

55.  93312  cu.  in. 

85.   1410'. 

pwts.  20  grs. 

56.  3328  cu.  ft. 

86.   147600". 

25.   12  Ibs.  8  oz. 

57.  2  cu.   ft.  774 

87.   14°. 

26.  4176  ounces. 

cu.  in. 

88.  Os.  16°  4-. 

27.  6200  Ibs. 

58.  25    cords,  64 

89.   216000". 

28.  2400  ounces. 

cu.  ft. 

90.   9120'. 

29.  62  Ibs.  8  oz. 

59.  756  pts. 

91,  0°  16'. 

30.  2  tons,  2  cwt.  60.  2200  gills. 

92.   1  sign.. 

2  qrs.  6  Ibs.  1  61.  2580  qts. 

PAGES  105 — 111.]  ANSWERS.  127 

REDUCTION  ASCENDING  AND  DESCENDING. 


Ex.      Ans. 

Ex.      Ans. 

Ex.      Ans. 

93.  45360  far. 
94.  £63. 
95.  248G.5s.8d. 
96.  80  G. 

104.  13r.l3f.  8  i. 
105.  69840  oz. 
106.  205554  grs. 
107.  36  E.  ells. 

116.  194  p.  Ik  43 
gals. 
117.  473353920s 
118.  13  wks.  1  d. 

97.  268440  grsr 
98.  143  1.  4  o.  1 

108.  56  yds.  1  qr. 
109.  72.  Fl.  ells. 

22  hrs.  20  min. 
119.  3581793s.  ft. 

p.  6g. 
99.  357360  grs. 
100.  33109  Ibs. 

110.  40  F.  ells. 
111.  839599  in. 
112.  5  1.  306  r.  7  f. 

120.  39-  ft.  1208  i. 
121.  2222208  c.in. 
122.  17003520  in. 

101.  24  T.  9  cwt. 
10  Ibs.  8  oz. 
102.  3682  in. 
103.  58278  ft. 

113.  8693  pts. 
114.  165  qrs.2bu. 
2  pks.  5  qts. 
115.  97344  gills. 

123.  13  A.  75  r. 
11^  yds. 
124.  1018818  sec. 
125.  418.24°,  20'. 

COMPOUND  ADDITION. 


5.  £40,  14s.2d.2f. 

9.  35  w.  4h.  21m. 

13.  6  pi.   18  gals. 

6.  59  1.  2  p.  22  g. 

10.  23  yds.  3  na. 

3  qts.  2  gi. 

7.  22  r.  1  yd.  5  in. 

11.  13T.12c.lqr. 

14.  37  bu.  3  pks. 

8.  26  cwt.  3  qrs.  5 

10  1.  9  o.  13  d. 

3  qts. 

Ibs.  5  oz. 

12.  27hhds.  38  g. 

15.  67  y.  3  q.  2  na. 

COMPOUND  SUBTRACTION. 

2.  £7,  Is.  9d.  2  far. 

6.  7  yds.  3  qrs.  1  n. 

12.  4  A.  2  roods,  4 

3.  3   Ibs.   7   oz.    4 

7.  £9,  17s.  4d. 

rods. 

pwts.  8  grs. 

8.  44  gals.  1  pt. 

13.  8  lea.  2  mi.  0 

4.  4  T.   17   cwt.  1 

9.  4T.  16  c.  74  Ib. 

fur.  4  r. 

qr.  24  Ibs. 

10.  2y.  3  mo.  16  d. 

14.  17  bu.  5  q.  2  p. 

5.  5   m.  5  fur.  7  r. 

11.  15  y.  10  mo.  3 

15.  45  G.  18s.  8d 

3  ft.  9  in. 

d.  8  h.  4  m. 

2  far. 

COMPOUND  MULTIPLICATION. 

1.  Given. 

5.  661.  285  r.  11  f. 

9.  625  y.  2  q.  2  n 

2.  £64,   13s.  5d.  1 

3  i. 

10.  2173  d.  13  h.  3 

farthing. 

6.  199  T.  14  c.  14 

m.  47  s. 

3.  236  1.5  o.  11  p. 

1.  15  o.  1  I. 

11.  2272   y.  30  w 

12  g. 

7.  43A.16r.84f  f. 

3  d.  12  h. 

4,  34  mi.  2  f.  20  r. 

8.  10  cords,  61  c.  f. 

12.  4707  h.  18  g. 

128 


ANSWERS.          [PAGES  113 — 118. 
COMPOUND  DIVISION. 


Ex.               A  us. 

Ex.               Ans. 

Ex.               Ans. 

1.  Given. 
2.  4  1.  9  oz.   15  p. 
11  g. 
3,  £1,  9s.  5d.  0£  f. 
4   51.  13  p.  15f  g. 
5.  2  T.  12   c.  2  q. 
14*1 

6.  3  y.  1  q.  If  na. 
7.  4    1.    1    in.   2  f. 
I7i  r. 
8.  5h.2g.4q.0-fp. 
9.  3  b.  3  p.  3f  q. 
10.  £5,  3s.  6d.  2  f. 
11.2  T.5  c.  3-H-l- 

12.  1  hhd.  49  gals. 
3|f  qts. 
13.  1    A.   2   rx>ds, 
l**r. 
14.  91  c.  f.  OH  i- 
15.  1    yr.    12  wks, 
4«  d. 

MISCELLANEOUS  EXERCISES. 


1.  271557. 

23.  460|f. 

51.  $279. 

76.  150  ap. 

2.  1229930. 

24.  17-ftV 

52.  $1180. 

77.  112  mar. 

3.  875431. 

25.  18-ftV 

53.  3399  s. 

78.  192  par. 

4.  372713- 

26.  12HH- 

54.  $90. 

79.  371  fans. 

124. 

27.  43i+H- 

55.  87  s. 

80.  25  2|  yds. 

5.  837598- 

28.  74VL(/Vi8V 

56.  13-f-g  yds. 

81.  512  pair. 

35. 

29.  261. 

57.  13-/g-  yds. 

82.  10962d. 

6.  24064H- 

30.  568280. 

58.  33-ft-  a. 

83.  60850  f. 

7.  1593i££. 

33.  $19. 

59.  20  cows. 

84.  302  yds. 

8.  245948, 

34.  $13. 

60.  25  bar. 

85.  4000. 

616  rem. 

35.  $17. 

61.  29  or. 

86.  1890  s. 

9.  4-f* 

36.  $49. 

62.  64  quarts. 

87.  $20. 

10.  6800. 

37.  86  Ibs. 

63.  60  Ibs. 

88.  7560  bot 

11.  98040. 

38.  63  s. 

64.  75  c. 

89.  $40.80. 

12.  53f£. 

39.  356  Ibs. 

65.  $108. 

90.  $124.16 

13.  108332. 

40.  $28. 

66,  $7.92. 

91.  $1100. 

14.  11542. 

41.  300  m. 

67.  $252. 

92.  $442. 

15.  33611. 

42.  $19. 

68.  936  m. 

93.  $3180. 

16.  26869. 

43.  $18. 

69.  45  cents. 

94.  $9600. 

17.  3810225. 

44.  $75. 

70.  $30. 

95-  $17280. 

18.  12469. 

45.  106  sk 

71.  $140. 

96.  $1981980. 

19.  2720325. 

46.  $837. 

72.  $150. 

97.  $960. 

20.  17. 

47.  $2008.  • 

73.  12  dress's. 

98.  $130.56. 

21.  5  and  9  r. 

48.  $1744. 

74.  16  boxes. 

99.  52  y.  6  m. 

22.  18  and 

49.  8114. 

75.  110  miles. 

1^0.  4  weeks, 

23.  424  over. 

50.:$165. 

- 

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